Dijkstra's algorithm, published in 1959 and named after its creator Dutch computer scientist Edsger Dijkstra, can be applied on a weighted graph. Very nice article. Graph Algorithms I 12. We use n = jV(G)jand m = jE(G)jto denote the number of ver-. Shortest Path Queries A shortest path query on a(n) (undirected) graph ﬁnds the shortest path for the given source and target vertices in the graph. The graph is not weighted. Fast Paths allows a massive speed-up when calculating shortest paths on a weighted directed graph compared to the standard Dijkstra algorithm. it, daniele. In the first part of the paper, we reexamine the all-pairsshortest paths (APSP) problem and present a newalgorithm with running time approaching O(n3/log2n), which improves all known algorithms for general real-weighted dense graphs andis perhaps close to the best result possible without using fast matrix multiplication, modulo a few log log n factors. Since the edges in the center of the graph have large weights, the shortest path between nodes 3 and 8 goes around the boundary of the graph where the edge weights are smallest. ! But what if edges have different 'costs'? s v δ(, ) 3sv = δ(, ) 12sv = 2 s v 2 5 1 7. Consumes a graph and two vertices, and returns the shortest path (in terms of number of vertices) between the two vertices. When There Is An Edge Between I And J, Then G[i][j] Denotes The Weight Of The Edge. Weighted graphs are commonly used in determining the most optimal path, most expedient, or the. I maintain a count for the number of shortest paths; I would like to use BFS from v first and also maintain a global level. Weighted Graph ( भारित ग्राफ ) Discrete Mathematics Shortest Path || Dijkstra Algorithm #weightedgraph #grewalpinky B. Shortest Path (Unweighted Graph) Goal: find the shortest route to go from one node to another in a graph. Exercise 3 [Modeling a problem as a shortest path problem in graphs] Four imprudent walkers are caught in the storm and nights. It maintains a set of nodes for which the shortest paths are known. 7 s: source 8 ’’’ 9 result = ShortestPathResult() 10 num_vertices = graph. the first assumes that the graph is weighted, which means that each edge has a cost to traverse it. BFS always visits nodes in increasing order of their distance from the source. An interesting problem is how to find shortest paths in a weighted graph; i. The previous state of the art for this problem was total update time O ̃(n 2 √m/ε) for directed, unweighted graphs [2], and Õ(mn=ε) for undirected, unweighted graphs [12]. The result is a list of vertices, or #f if there is no path. G Is A Weighted Graph With Vertex Set {0,1,2,,1-1} And Integer Weights Represented As The Following Adjacency Matrix Form. Assignment: Given any connected, weighted graph G, use Dijkstras algorithm to compute the shortest (or smallest weight) path from any vertex a to any other vertex b in the graph G. Our main goal is to characterize exactly which sets of node sequences, which we call path systems, can appear as unique shortest paths in a graph with arbitrary real edge weights. If There Is An Edge Between I And 1, Then G[16] > 0, Otherwise G[i,j]=-1. I'm using the networkx package in Python 2. If the graph is weighted (that is, G. Shortest paths. Here a, b, c. A cycle is a path where the first and last vertices are the same. We wish to determine a shortest path from v 0 to v n Dijkstra's Algorithm Dijkstra's algorithm is a common algorithm used to determine shortest path from a to z in a graph. Dijkstra's Algorithm. Counting the number of shortest paths in various graphs is an important and interesting combinatorial problem, especially in weighted graphs with various applications. A few months ago, mathematicians Andrew Beveridge and Jie Shan published Network of Thrones in Math Horizon Magazine where they analyzed a network of character interactions from the novel “A Storm of Swords”, the third book in the popular “A Song of Ice and Fire” and the basis for the Game of Thrones TV series. numPaths initialized to 1). Combinatorics is the branch of mathematics concerned with selecting, arranging, and listing or counting collections of objects. Newman Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501 and Center for Applied Mathematics, Cornell University, Rhodes Hall, Ithaca, New York 14853 ~Received 1 February 2001; published 28 June 2001!. Shortest Path In A Weighted Directed Graph With Dijkstra's Algorithm - posted in C and C++: Well, I encountered an interesting problem. We start with undirected graphs. The main algorithms that fall under this definition are Breadth-First Search (BFS) and Dijkstra's algorithms. A logical scalar. Weighted Graphs Data Structures & Algorithms 2 [email protected] ©2000-2009 McQuain Shortest Paths (SSAD) Given a weighted graph, and a designated node S, we would like to find a path of least total weight from S to each of the other vertices in the graph. The graph given in the test case is shown as : The shortest paths for the 3 queries are :: The direct Path is shortest with weight 5: There is no way of reaching node 1 from node 3. It is used to identify optimal driving directions or degree of separation between two people on a social network for example. The gist of Dijkstra's single source shortest path algorithm is as below : Dijkstra's algorithm finds the shortest path in a weighted graph containing only positive edge weights from a single source. The shortest path between two points in a weighted graph can be found with Dijkstra’s algorithm. Now we are going to find the shortest path between source (a) and remaining vertices. Find the shortest distance from C to D and if it is impossible to reach node D from C then return -1. The algorithm exists in many variants. This means that, given a weighted graph, this algorithm will output the shortest distance from a selected node to all other nodes. shortest path. In this post, I explain the single-source shortest paths problems out of the shortest paths problems, in which we need to find all the paths from one starting vertex to all other vertices. Changing to its dual, the triangular grid, paths between triangle pixels (we abbreviate this term to trixels) are counted. Question: Shortest Paths(int[0). The total cost of a path in a graph is equal to the sum of the weights of the edges that connect the vertices in the path. implementations of the shortest-path search algorithms on graphs on the number of graph vertices experimentally. In the first part of the paper, we reexamine the all-pairsshortest paths (APSP) problem and present a newalgorithm with running time approaching O(n3/log2n), which improves all known algorithms for general real-weighted dense graphs andis perhaps close to the best result possible without using fast matrix multiplication, modulo a few log log n factors. PRACTICE PROBLEM BASED ON FLOYD WARSHALL ALGORITHM- Problem- Consider the following directed weighted graph- Using Floyd Warshall Algorithm, find the shortest path distance between every pair of vertices. The shortest path from 0 to 4 uses the shortest path from 0 to 1 and the edge 1–4. Our algorithm is deterministic and has a running time of O(k(m√n + n 3/2 log n)) where m is the number of edges in the graph and n is the number of vertices. In this graph, vertex A and C are connected by two parallel edges having weight 10 and 12 respectively. Abstract: In this paper we evaluate our presented Quantum Approach for finding the Estimation of the Length of the Shortest Path in a Connected Weighted Graph which is achieved with a polynomial time complexity about O(n) and as a result of evaluation we show that the Probability of Success of our presented Quantum Approach is increased if the Standard Deviation of the Length of all possible. An assigned number is called the weight of the edge, and the collection of all weights is called a weighting of the graph Γ. A path in a graph is a sequence of adjacent vertices. In any graph G, the shortest path from a source vertex to a destination vertex can be calculated using Dijkstra Algorithm. Path Graphs. Only edges with non-negative costs are included. Discuss an efficient algorithm to compute a shortest path from node s to node t in a weighted directed graph G such that the path is of minimum cardinality among all shortest s - t paths in G graph-theory hamiltonian-path path-connected. This means it finds a shortest paths between nodes in a graph, which may represent, for example, road networks; For a given source node in the graph, the algorithm finds the shortest path between source node and every other node. dijkstra_path¶ dijkstra_path (G, source, target, weight='weight') [source] ¶. 0 = no epidemic 1 = epidemic NW N SW C E Graph 3: Vertex coloring of weighted graph for the county. In this category, Dijkstra’s algorithm is the most well known. You are expected to do it in Time Complexity of O(A + M). shortest path functions use it as the cost of the path; community finding methods use it as the strength of the relationship between two vertices, etc. Shortest path algorithms are a family of algorithms designed to solve the shortest path problem. The shortest path between node 222 and node 444 is 222 -> 555 -> 666 -> 777 -> 444, which has a weighted distance 1. The number of diagonal steps in a shortest path of the chessboard distance is min{w 1,w 2}, and the number of cityblock steps (i. A cycle is a path where the first and last vertices are the same. Given an undirected, weighted graph, find the minimum number of edges to travel from node 1 to every other node. (Consider what this means in terms of the graph shown above right. The Edge can have weight or cost associate with it. Simple path is a path with no repeated vertices. Hence, parallel computing must be applied. In this post I will be discussing two ways of finding all paths between a source node and a destination node in a graph: Using DFS: The idea is to do Depth First Traversal of given directed graph. , capacity, cost, demand, traffic frequency, time, etc. There are already comprehensive network analysis packages in R, notably igraph and its tidyverse-compatible interface tidygraph. G Is A Weighted Graph With Vertex Set {0,1,2,,1-1} And Integer Weights Represented As The Following Adjacency Matrix Form. Geodesic paths are not necessarily unique, but the geodesic distance is well-defined since all geodesic paths have. ple, Figure 1a illustrates a graph G, and Figure 1e shows an aug-mented graph G∗ constructed from G. The latter only works if the edge weights are non-negative. 4 5 Args: 6 graph: weighted graph with no negative cycles. Weighted directed graphs may be used to model communication networks, and shortest distances (shortest-path weights) between nodes may be used to suggest routes for messages. Chan⁄ September 30, 2009 Abstract Intheﬂrstpartofthepaper,wereexaminetheall-pairs shortest paths (APSP)problemand present a new algorithm with running time O(n3 log3 logn=log2 n), which improves all known algorithmsforgeneralreal-weighteddensegraphs. This module covers weighted graphs, where each edge has an associated weight or number. The problem is to find k directed paths starting at s, such that every node of G lies on at least one of those paths, and such that the sum of the weights of all the edges in the paths is minimized. The one-to-all shortest path problem is the problem of determining the shortest path from node s to all the other nodes in the. Finding shortest paths in weighted graphs In the past two weeks, you've developed a strong understanding of how to design classes to represent a graph and how to use a graph to represent a map. Exercise 3 [Modeling a problem as a shortest path problem in graphs] Four imprudent walkers are caught in the storm and nights. How to use BFS for Weighted Graph to find shortest paths ? If your graph is weighted, then BFS may not yield the shortest weight paths. You are given a weighted directed acyclic graph G, and a start node s in G. It finds a shortest-path tree for a weighted undirected graph. Journal of the ACM 65 :6, 1-40. Is there a cycle that uses each vertex. Mizrahi et al. The number dist[w] equals the length of a shortest path from v to w, or is -1 if w cannot be reached. Dijkstra's Algorithm is useful for finding the shortest path in a weighted graph. Given a directed weighted graph G= (V;E;w) with non-negative weights w: E!R+ and a vertex s2V, the single-source shortest paths is the family of shortest paths s vfor every vertex v2V. If finds only the lengths not the path. If There Is An Edge Between I And 1, Then G[16] > 0, Otherwise G[i,j]=-1. A logical scalar. We use n = jV(G)jand m = jE(G)jto denote the number of ver-. The order of a graph is the number of nodes. When There Is An Edge Between I And J, Then G[i][j] Denotes The Weight Of The Edge. The Dijkstra’s algorithm is provided in Algorithm 2. For example, the length of v8,v9 equals 2, which is identical to the length of the. As such, we say that the weight of a path is the sum of the weights of the edges it contains. The shortest-path from vertex u to vertex v in a weighted graph is a path with minimum sum-weight B All-Pairs Shortest Paths For a weighted graph, the all-pairs shortest paths problem is to nd the shortest path between all pairs of vertices. 2 Representing Weighted Graphs Often it is desirable to use a priority queue to store weighted edges. Question: Shortest Paths(int[0). We first propose an exact (and. A single execution of the algorithm will. shortest path. Consider a shortest path p from vertex i to vertex j, and suppose that p contains at most m edges. In the weighted matching [G85b, GT89, GT91] and maxi-mum ﬂow problems [GR98], for instance, the best algorithms for real- and integer-weighted graphs have running times diﬀering by a polynomial factor. Slight Adjustment of Dijkstra's Algorithm to Solve Shortest Path Problem… 783 The i-number weight of an edge is also called the i-distance between the corresponding two nodes. The distance between two vertices u and v, denoted d(u,v), is the length of a shortest. Suppose you are given a directed graph G = (V, E), with costs on the edges; the costs may be. A path in a graph is a sequence of adjacent vertices. The previous state of the art for this problem was total update time O ̃(n 2 √m/ε) for directed, unweighted graphs [2], and Õ(mn=ε) for undirected, unweighted graphs [12]. Assignment: Given any connected, weighted graph G, use Dijkstras algorithm to compute the shortest (or smallest weight) path from any vertex a to any other vertex b in the graph G. Shortest Path. An example of a weighted graph would be the distance between the capitals of a set of countries. You are given a weighted undirected graph. Design a linear time algorithm to find the number of different shortest paths (not necessarily vertex disjoint) between v and w. We present an improved algorithm for maintaining all-pairs (1 + ε) approximate shortest paths under deletions and weight-increases. Consider the graph above. Your task is to find the shortest path between the vertex 1 and the vertex n. In weighted graphs, where we assume that the edge weights do not represent the communication speed, a straightforward distributed variant of the Bellman-Ford algorithm [2], [3], [4] computes. Shortest paths problems are among the most fundamental algorithmic graph problems. The actual shortest paths can also be constructed by modifying the above algorithm. Question: Shortest Paths(int[0). If There Is An Edge Between I And 1, Then G[16] > 0, Otherwise G[i,j]=-1. To find the shortest path on a weighted graph, just doing a breadth-first search isn't enough - the BFS is only a measure of the shortest path based on number of edges. A comparison of the data obtained as a result of the study was carried out to find the best applications of implementations of the shortest path search algorithms in the Postgre SQL DBMS. Single source shortest path for undirected graph is basically the breadth first traversal of the graph. Shortest Path Algorithms?(a) Breadth-first search (BFS) can be used to perform single source short- est paths on any graph where all edges have the same costs 1. A cycle is a path where the first and last vertices are the same. Let s denote the number of edges of H. Shortest-Path Problems (cont'd) Single-source shortest path problem Given a weighted graph G = (V, E), and a distinguished start vertex, s, find the minimum weighted path from s to every other vertex in G The shortest weighted path from v 1 to v 6 has a cost of 6 and v 1 v 4 v 7 v 6. Shortest Path Problem. Dijkstra's algorithm (or Dijkstra's Shortest Path First algorithm, SPF algorithm) is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. Node is a vertex in the graph at a position. Assignment: Given any connected, weighted graph G, use Dijkstras algorithm to compute the shortest (or smallest weight) path from any vertex a to any other vertex b in the graph G. Using similar ideas, we can construct a (1+epsilon)-approximate distance oracle for weighted unit-disk graphs with O(1) query. Finding the shortest path, with a little help from Dijkstra! If you spend enough time reading about programming or computer science, there’s a good chance that you’ll encounter the same ideas. in Proceedings - 25th IEEE International Conference on High Performance Computing, HiPC 2018. Shortest Path Problem. Graph Algorithms I 12. We let k denote the number of source. Given a weighted line-graph (undirected connected graph, all vertices of degree 2, except two endpoints which have degree 1), devise an algorithm that preprocesses the graph in linear time and can return the distance of the shortest path between any two vertices in constant time. The All-Pairs Shortest Paths (APSP) problem seeks the shortest path distances between all pairs of vertices, and is one of the most fundamental graph problems. Once we have reached our destination, we continue searching until all possible paths are greater than 11; at that point we are certain that the shortest path is 11. The vertices V are connected to each other by these edges E. Key Graph Based Shortest Path Algorithms With Illustrations - Part 1: Dijkstra's And Bellman-Ford Algorithms Bellman-Ford algorithm is used to find the shortest paths from a source vertex to all other vertices in a weighted graph. Distributed Exact Weighted All-Pairs Shortest Paths in O˜(n5/4) network is modeled by a weighted n-node m-edge graph G. There are already comprehensive network analysis packages in R, notably igraph and its tidyverse-compatible interface tidygraph. We revisit a classical graph-theoretic problem, the single-source shortest-path (SSSP) problem, in weighted unit-disk graphs. An interesting side-effect of traversing a graph in BFS order is the fact that, when we visit a particular node, we can easily find a path from the source node to the newly visited node with the least number of edges. The Symmetric Shortest-Path Table Routing Conjecture Thomas L. Question: Shortest Paths(int[0). Mizrahi et al. C++ Program to Generate a Random UnDirected Graph for a Given Number of Edges; Shortest Path in a Directed Acyclic Graph. Shortest Distance in a graph with two different weights : Given a weighted undirected graph having A nodes, a source node C and destination node D. Exercise 7 Consider the following modification of the Dijkstra’s algorithm to work with negative weights:. The algorithm creates a tree of shortest paths from the starting vertex, the source, to all other points in the graph. The previous state of the art for this problem was total update time O ̃(n 2 √m/ε) for directed, unweighted graphs [2], and Õ(mn=ε) for undirected, unweighted graphs [12]. I will cover several other graph based shortest path algorithms with concrete illustrations. Shortest path in a graph with weighted edges and vertices. If there are multiple paths from node 1 to a node that have the same minimum number of. Hi I have already posted a similar question but there was a misunderstanding of the problem by my side so here I post it again. There is one shortest path vertex 0 to vertex 0 (from each vertex there is a single shortest path to itself), one shortest path between vertex 0 to vertex 2 (0->2), and there are 4 different shortest paths from vertex 0 to vertex 6: 1. A shortest path between two nodes u and v in a graph is a path that starts at u and ends at v and has the lowest total link weight. Given an undirected, weighted graph, find the minimum number of edges to travel from node 1 to every other node. Row i of the predecessor matrix contains information on the shortest paths from point i: each entry predecessors[i, j] gives the index of the previous node in the path from point i to point j. You are expected to do it in Time Complexity of O(A + M). For example, the length of v8,v9 equals 2, which is identical to the length of the. A shortest path (with the lowest weight) from $$u$$ to $$v$$; The weight of the path is the sum of the weights of its edges. Both algorithms were randomized and had constant query time. It was conceived by computer scientist Edsger W. Deterministic Partially Dynamic Single Source Shortest Paths in Weighted Graphs Aaron Bernstein May 30, 2017 Abstract In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph Gand a source node sthe goal is to maintain shortest distances between sand all other nodes. In this category, Dijkstra's algorithm is the most well known. An undirected graph that has a path from every vertex to every other vertex in the graph. As noted earlier, mapping software like Google or Apple maps makes use of shortest path algorithms. C++ Program to Generate a Random UnDirected Graph for a Given Number of Edges; Shortest Path in a Directed Acyclic Graph. If you think carefully, it's easy to see that there can be many graphs such that the. A new approach to all-pairs shortest paths on real-weighted graphs Seth Pettie1 Department of Computer Sciences, The University of Texas at Austin, Austin, TX 78712, USA Abstract We present a new all-pairs shortest path algorithm that works with real-weighted graphs in the traditional comparison-additionmodel. The gist of Dijkstra's single source shortest path algorithm is as below : Dijkstra's algorithm finds the shortest path in a weighted graph containing only positive edge weights from a single source. How to use BFS for Weighted Graph to find shortest paths ? If your graph is weighted, then BFS may not yield the shortest weight paths. Chan⁄ September 30, 2009 Abstract Intheﬂrstpartofthepaper,wereexaminetheall-pairs shortest paths (APSP)problemand present a new algorithm with running time O(n3 log3 logn=log2 n), which improves all known algorithmsforgeneralreal-weighteddensegraphs. shortest path algorithm. In this tutorial, we will present a general explanation of both algorithms. A cycle is a path where the first and last vertices are the same. techniques to speed the computation of shortest paths in the discretization graph [3,21]. ; It uses a priority based dictionary or a queue to select a node / vertex nearest to the source that has not been edge relaxed. This turns out to be a problem that can be solved efficiently, subject to some restrictions on the edge costs. An observed image is composed of multiple components based on optical phenomena, such as light reflection and scattering. The multiplicity of a path is the maximum number of times that an edge appears in it. The latter only works if the edge weights are non-negative. It maintains a set S of vertices whose final shortest path from the source has already been determined and it repeatedly selects the left vertices with the minimum shortest-path estimate, inserts them. Fast Paths allows a massive speed-up when calculating shortest paths on a weighted directed graph compared to the standard Dijkstra algorithm. Directed and undirected graphs may both be weighted. I maintain a count for the number of shortest paths; I would like to use BFS from v first and also maintain a global level. Single Source Shortest Path. This section discusses three algorithms for this problem: breadth-ﬁrst search for unweighted graphs, Dijkstra’s algorithm for weighted graphs, and the Floyd-Warshall algorithm for computing distances between all pairs of vertices. And here is some test code: test_graph. Weighted directed graphs may be used to model communication networks, and shortest distances (shortest-path weights) between nodes may be used to suggest routes for messages. An undirected graph that has a path from every vertex to every other vertex in the graph. shortest path problem. shortest_paths calculates a single shortest path (i. We will see how simple algorithms like depth-ﬁrst-search can be used in clever ways (for a problem known as topological sorting) and will see how Dynamic Programming can be used to solve problems of ﬁnding shortest paths. yenpathy: An R Package to Quickly Find K Shortest Paths Through a Weighted Graph Submitted 05 September 2019 This paper is under review which means review has begun. ! Here, the length of a path is simply the number of edges on the path. One of these challenges is the need to generate a limited number of test cases of a given regression test suite in a manner that does not compromise its defect detection ability. For example ﬁnding the ‘shortest path’ between two nodes, e. We would then assign weights to vertices, not edges. Python – Get the shortest path in a weighted graph – Dijkstra Posted on July 22, 2015 by Vitosh Posted in VBA \ Excel Today, I will take a look at a problem, similar to the one here. FindShortestPath[g, s, All] generates a ShortestPathFunction[] that can be applied repeatedly to different t. [email protected] ple, Figure 1a illustrates a graph G, and Figure 1e shows an aug-mented graph G∗ constructed from G. According to [11], [12], [13], since 1959 Dijkstra's algorithm has been recognized as the best algorithm and used as method to find the shortest path. nodes in a given directed graph is a very common problem. You are also given a positive integer k. A subgraph is a subset of a graph’s edges (with associated vertices) that form a graph. ” Dijkstra’s algorithm is an iterative algorithm that provides us with the shortest path from one particular starting node to all other nodes in the graph. The shortest path function can also be used to compute a transitive closure or for arbitrary length traversals. the number of pairs of vertices not including v, which for directed graphs is and for undirected graphs is. The all-pairs shortest paths problem for unweighted directed graphs was introduced by Shimbel (1953), who observed that it could be solved by a linear number of matrix multiplications that takes a total time of O(V 4). There are two types of weighted graphs: vertex weighted and edge weighted. Shortest path problem (SPP) is a fundamental and well-known combinatorial optimization problem in the area of graph theory. A weighted graph is a one which consists of a set of vertices V and a set of edges E. The shortest path problem is about finding a path between $$2$$ vertices in a graph such that the total sum of the edges weights is minimum. Dijkstra's Shortest Path Algorithm in Java. The single pair shortest path problem seeks to compute (u;v) and construct a shortest path from. Consumes a graph and two vertices, and returns the shortest path (in terms of number of vertices) between the two vertices. A subgraph is a subset of a graph’s edges (with associated vertices) that form a graph. If There Is An Edge Between I And 1, Then G[16] > 0, Otherwise G[i,j]=-1. the sum of the weights of the edges in the paths is minimized. shortest path algorithm. Variations of the Shortest Path Problem. We present a new all-pairs shortest path algorithm that works with real-weighted graphs in the traditional comparison-addition model. I Therefore, the numbers d 1;d 2; ;d n must include an even number of odd numbers. Your solution should be complete in that it shows the shortest path from all starting vertices to all other vertices. All-Pairs Shortest Paths Problem To ﬁnd the shortest path between all verticesv 2 V for a graph G =(V,E). This algorithm has numerous applications in network analysis, such as transportation planning. (2018) Decremental Single-Source Shortest Paths on Undirected Graphs in Near-Linear Total Update Time. Dijkstra’s Shortest Path Algorithm in Java. Fast Paths allows a massive speed-up when calculating shortest paths on a weighted directed graph compared to the standard Dijkstra algorithm. It was conceived by computer scientist Edsger W. Shortest Path in a weighted Graph where weight of an edge is 1 or 2 Given a directed graph where every edge has weight as either 1 or 2, find the shortest path from a given source vertex ‘s’ to a given destination vertex ‘t’. Given an undirected, weighted graph, find the minimum number of edges to travel from node 1 to every other node. be contained in shortest augmenting paths, and the lay-ered network contains all augmenting paths of shortest length. Dating back some 3000 years, and initially consisting mainly of the study of permutations and combinations, its scope has broadened to include topics such as graph theory, partitions of numbers, block designs, design of codes, and latin squares. Simple path is a path with no repeated vertices. Find a TSP solution using state-of-the-art software, and then remove that dummy node (subtracting 2 from the total weight). C… Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. This module covers weighted graphs, where each edge has an associated weight or number. Discuss an efficient algorithm to compute a shortest path from node s to node t in a weighted directed graph G such that the path is of minimum cardinality among all shortest s - t paths in G graph-theory hamiltonian-path path-connected. Dijkstra’s Algorithms describes how to find the shortest path from one node to another node in a directed weighted graph. Print the number of shortest paths from a given vertex to each of the vertices. Given a weighted line-graph (undirected connected graph, all vertices of degree 2, except two endpoints which have degree 1), devise an algorithm that preprocesses the graph in linear time and can return the distance of the shortest path between any two vertices in constant time. Dijkstra's Algorithm: Finds the shortest path from one node to all other nodes in a weighted graph. It finds a shortest-path tree for a weighted undirected graph. The SQL Server graph extensions are amazing. Introduction. In this week, you'll add a key feature of map data to our graph representation -- distances -- by adding weights to your edges to produce a "weighted. Must Read: C Program. The Degree of a vertex is the number of edges incident on it. I'm using the networkx package in Python 2. adjacent b. Shortest Paths: Problem Statement Given a weighted graph and two vertices u and v, we want to find a path of minimum total weight between u and v Length (or distance) of a path is the sum of the weights of its edgesLength (or distance) of a path is the sum of the weights of its edges. Floyd-Warshall algorithm is an algorithm for finding shortest paths in a weighted graph with positive or negative edge weights (but with no negative cycles). The Degree of a vertex is the number of edges incident on it. for unweighted. If the graph is unweighed, then finding the shortest path is easy: we can use the breadth-first search algorithm. A path graph is a graph consisting of a single path. So, the shortest path would be of length 1 and BFS would correctly find this for us. Slight Adjustment of Dijkstra's Algorithm to Solve Shortest Path Problem… 783 The i-number weight of an edge is also called the i-distance between the corresponding two nodes. We know that breadth-first search can be used to find shortest path in an unweighted graph or in weighted graph having same cost of all its edges. Shortest path in a graph with weighted edges and vertices. PY - 2007/10/30. If There Is An Edge Between I And 1, Then G[16] > 0, Otherwise G[i,j]=-1. Variations of the Shortest Path Problem. We start with a directed and weighted graph $$G = (V, E)$$ with a weights function $$w: E \rightarrow R$$ that maps edges to weights with real values; Find for each pair of vertices $$u$$, $$v \in V$$. Shortest paths&Weighted graphs. An undirected graph that has a path from every vertex to every other vertex in the graph. Consider any node that is not the root: its possible distances from the root are all possible distances of its neighbors plus the weight of the connecting edges. If an edge is missing a special value, perhaps a negative value, zero or a large value to represent "infinity", indicates this fact. In a weighted graph with start node s, there are often multiple shortest paths from s to any other node. 1 def shortest_path_cycle(graph, s): 2 ’’’Single source shortest paths using DP on a graph with cycles but no 3 negative cycles. Dijkstra’s algorithm is similar to Prim’s algorithm. 4 5 Args: 6 graph: weighted graph with no negative cycles. b) Weighted graph matching: Mulmuley & Shah ob-served that their result for the shortest path problem yields the same lower bound for the WEIGHTED GRAPH MATCHING problem [3, Corollary 1. In any graph G, the shortest path from a source vertex to a destination vertex can be calculated using this algorithm. 1 Shortest paths and matrix multiplication 25. This means it finds a shortest paths between nodes in a graph, which may represent, for example, road networks; For a given source node in the graph, the algorithm finds the shortest path between source node and every other node. We know that breadth-first search can be used to find shortest path in an unweighted graph or in weighted graph having same cost of all its edges. Suppose we have to following graph: We may want to find out what the shortest way is to get from node A to node F. Graphs: Finding shortest paths Definition: weight of a path 4 Tecniche di programmazione A. Weighted Graph ( भारित ग्राफ ) Discrete Mathematics Shortest Path || Dijkstra Algorithm #weightedgraph #grewalpinky B. P = shortestpath(G,s,t) computes the shortest path starting at source node s and ending at target node t. 18 between shortest paths in networks and Hamilton paths in graphs ties in with our observation that finding paths of low weight (which we have been calling "short") is tantamount to finding paths with a high number of edges (which we might consider to be "long"). Exercise 7 Consider the following modification of the Dijkstra’s algorithm to work with negative weights:. If you think carefully, it's easy to see that there can be many graphs such that the. A path in a graph is a sequence of adjacent vertices. As there are a number of different shortest path algorithms, we've gathered the most important to help you understand how they work and which is the best. This problem also is known as "Print all paths between two nodes". The shortest paths followed for the three nodes 2, 3 and 4 are as follows : 1/S->2 - Shortest Path Value : 1/S->3 - Shortest Path Value : 1/S->3->4 - Shortest Path Value :. Data Structure Graph Algorithms Algorithms. It runs in O (mn+n 2 log log n) time, improving on the long-standing bound of O (mn+n 2 log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps. The total cost of a path in a graph is equal to the sum of the weights of the edges that connect the vertices in the path. By contrast, the graph you might create to specify the shortest path to hike every trail could be a directed graph, where the order and direction of edges matters. 6 2, 6(a), 6(c), 18 In Exercises 2-4 find the length of a shortest path between a and z in the given weighted graph. Shortest Paths, and Dijkstra's Algorithm: Overview Graphs with lengths/weights/costs on edges. Scientiﬁc collaboration networks. 2 You can use the tool to create a weighted graph with mouse gestures and show the MST and shortest paths. If say we were to find the shortest path from the node A to B in the undirected version of the graph, then the shortest path would be the direct link between A and B. shortest_paths calculates a single shortest path (i. Weighted graphs and path length Weighted graphs A weighted graph is a graph whose edges have weights. Topics in this lecture include:. You may print the results of this algorithm to the screen or to a log file. A B C F D E 7 14 9 8 15 2 9 4 6 Figure 3. Dijkstra’s algorithm is similar to Prim’s algorithm. This module covers weighted graphs, where each edge has an associated weight or number. I'm restricting myself to Unweighted Graph only. The relationship shown in the proof of Property 21. Shortest Path (Unweighted Graph) Goal: find the shortest route to go from one node to another in a graph. Undirected graph. Shortest paths in an edge-weighted digraph 4->5 0. This means it finds the shortest paths between nodes in a graph, which may represent, for example, road networks; For a given source node in the graph, the algorithm finds the shortest path between the source node and every other node. If the graph is weighted, it is a path with the minimum sum of edge weights. Algorithms to find shortest paths in a graph are given later. Breadth First Search, BFS, can find the shortest path in a non-weighted graphs or in a weighted graph if all edges have the same non-negative weight. More Algorithms for All-Pairs Shortest Paths in Weighted Graphs Timothy M. world as a directed graph, the problem of directing the robot was transformed into a shortest-path problem. As noted earlier, mapping software like Google or Apple maps makes use of shortest path algorithms. Variations of the Shortest Path Problem. If an edge is missing a special value, perhaps a negative value, zero or a large value to represent "infinity", indicates this fact. Print the number of shortest paths from a given vertex to each of the vertices. Consider the graph above. The best algorithm (modulo small polylogarithmic improvements) for this problem runs in cubic time, a. By contrast, the graph you might create to specify the shortest path to hike every trail could be a directed graph, where the order and direction of edges matters. Your solution should be complete in that it …. Representing a Graph. For example, the two paths we mentioned in our example are C, B and C, A, B. Thanks for pointing to Gephi. In order to solve the load-balancing problem for coarse-grained parallelization, the relationship between the computing time of a single-source shortest-path length of node and the features of node is studied. The all-pairs shortest paths problem for unweighted directed graphs was introduced by Shimbel (1953), who observed that it could be solved by a linear number of matrix multiplications that takes a total time of O(V 4). CS 340 Programming Assignment VII: Single-Source Shortest Paths in generally Weighted graphs Description: You are to implement both the DAG SP algorithm and the Bellman-Ford algorithm for single-source shortest paths based on whether or not you detect a cycle. a i g f e d c b h 25 15 10 5 10. Hart, Nilsson, and Raphael [12] discovered how to use this information to improve the efficiency of computing the shortest path. When There Is An Edge Between I And J, Then G[i][j] Denotes The Weight Of The Edge. The number parent[w] is the predecessor of w on a shortest path from v to w, or -1 if none exists. minimum path sizeis the shortest distance measured in the number of edges traversed. The algorithm can be implemented with O(m+nlog(n)) time; therefore, it is efficient in sparsely connected networks. The Bellman-Ford algorithm is a single-source shortest path algorithm. Michael Quinn, Parallel Programming in C with MPI and OpenMP,. Decomposing the observed image into individual components is an important process for various computer vision tasks. Description and notations-building the graph Given a weighted graph G = (V, E), where V. Given an undirected, weighted graph, find the minimum number of edges to travel from node 1 to every other node. By comparison, if the graph is permitted. The order of a graph is the number of nodes. P( s,t ) is the shortest path between the given vertices and containing the least sum of edge weights on the path from to. Bellman-Ford Algorithm is computes the shortest paths from a single source vertex to all of the other vertices in a weighted digraph. Shortest paths&Weighted graphs. It was conceived by computer scientist Edsger W. Finding the shortest path, with a little help from Dijkstra! If you spend enough time reading about programming or computer science, there’s a good chance that you’ll encounter the same ideas. This technique is applied to a minute level bid/ask quote dataset consisting of rates constructed from all G10 currency pairs. • In addition, the first time we encounter a vertex may, we may not have found the shortest path to it, so we need to delay committing to that path. 1 Shortest Path Queries Let G= (V;E;˚) be a road network (i. Start by setting the distance of all notes to infinity and the source's distance to 0. The previous state of the art for this problem was total update time O ̃(n 2 √m/ε) for directed, unweighted graphs [2], and Õ(mn=ε) for undirected, unweighted graphs [12]. A path in a graph is a sequence of adjacent vertices. shortest s-tpath in G n. Algorithms to find shortest paths in a graph are given later. These weights represent the cost to traverse the edge. A cycle is a path where the first and last vertices are the same. In any graph G, the shortest path from a source vertex to a destination vertex can be calculated using Dijkstra Algorithm. It turns out that it is as easy to compute the shortest paths from s to every node in G (because if the shortest path from s to t is s = v0, v1, v2, , vk = t, then the path v0,v1 is the shortest path from s to v1, the path v0,v1,v2 is the shortest path from s to v2, the path v0,v1,v2,v3 is the shortest path from s to v3, etc. Assignment: Given any connected, weighted graph G, use Dijkstras algorithm to compute the shortest (or smallest weight) path from any vertex a to any other vertex b in the graph G. Essentially, you replace the stack used by DFS with a queue. I'm looking for an algorithm that will perform as in the title. Again this is similar to the results of a breadth first search. It can be used in numerous fields such as graph theory, game theory, and network. Dijkstra's Shortest Path Algorithm in Java. linear-time. So there is a weighted graph, and the shortest paths of all pair of n. dijkstra_path_length¶ dijkstra_path_length (G, source, target, weight='weight') [source] ¶. What is the longest simple path between s and t? Cycle. Graphs can be weighted (edges carry values) and directional (edges have direction). Let u and v be two vertices in G, and let P be a path in G from u to v. We are also given a starting node s ∈ V. Dijkstra's Single Source Shortest Path. Exercise 7 Consider the following modification of the Dijkstra’s algorithm to work with negative weights:. Among the various shortest path algorithms, Dijkstra’s shortest path algorithm [1] is said to have better performance with regard to run time than the other algorithms. Finally, at k = 4, all shortest paths are found. Shortest Path in a weighted Graph where weight of an edge is 1 or 2 Given a directed graph where every edge has weight as either 1 or 2, find the shortest path from a given source vertex ‘s’ to a given destination vertex ‘t’. However, the robot had a means for estimating Euclidean distances in his world. Weighted graphs and path length Weighted graphs A weighted graph is a graph whose edges have weights. The Floyd-Warshall algorithm is an example of dynamic programming. 2 commits 1 branch. The previous state of the art for this problem was total update time O ̃(n 2 √m/ε) for directed, unweighted graphs [2], and Õ(mn=ε) for undirected, unweighted graphs [12]. The Floyd Warshall Algorithm is for solving the All Pairs Shortest Path problem. However, most communication networks are dynamic, i. shortest path problem. Referred to as the shortest path between vertices For weighted graphs this is the path that has the smallest sum of its edge weights ijkstra’salgorithm finds the shortest path between one vertex and all other vertices The algorithm is named after its discoverer, Edgser Dijkstra 24 The shortest path between B and G is: 1 4 3 5 8 2 2 1 5 1 B A. Design a linear time algorithm to find the number of different shortest paths (not necessarily vertex disjoint) between v and w. Graph Characteristics-Undirected-Weighted-Journey: (1, 7)-Shortest path: 1 – 4 – 6 – 7 (in purple)-Total cost: 6. We first propose an exact (and. The graph has eight nodes. The Degree of a vertex is the number of edges incident on it. For the shortest path problem on positively weighted graphs the integer/real gap is only logarith-mic. Fine the shortest weighted path from a vertex, s, to every other vertex in the graph. The weights of the edges can be positive or negative. A near linear shortest path algorithm for weighted undirected graphs Abstract: This paper presents an algorithm for Shortest Path Tree (SPT) problem. Here the graph we consider is unweighted and hence the shortest path would be the number of edges it takes to go from source to destination. The longest path is based on the number of edges in the path if weighted == false and the unweighted shortest path algorithm is being used. Your solution should be complete in that it shows the shortest path from all starting vertices to all other vertices. Suppose G be a weighted directed graph where a minimum labeled w(u, v) associated with each edge (u, v) in E, called weight of edge (u, v). We introduce a metric on the. It asks for the number of different shortest paths. Day 3: Weighted Graphs. This problem also is known as "Print all paths between two nodes". So there is a weighted graph, and the shortest paths of all pair of n. texens | Hello World ! | Page 3 Hello World !. In this tutorial, we will present a general explanation of both algorithms. The graph given in the test case is shown as : * The lines are weighted edges where weight denotes the length of the edge. Graphs can be weighted (edges carry values) and directional (edges have direction). We present an improved algorithm for maintaining all-pairs (1 + ε) approximate shortest paths under deletions and weight-increases. Give an efficient algorithm to solve the single-destination shortest paths problem. Thus, the shortest path between any two nodes is the path between the two nodes with the lowest total length. The weighted path length is given by ∑C(vi,vi+1) i=1 k-1 The general problem Given an edge-weighted graph G = (V,E) and two vertices, vs ∈V and vd ∈V, find the path that starts at vs and ends at vd that has the smallest weighted path length Single-source shortest. Uses Dijkstra's Method to compute the shortest weighted path between two nodes in a graph. It finds a shortest-path tree for a weighted undirected graph. Crossref, ISI, Google Scholar; 19. When There Is An Edge Between I And J, Then G[i][j] Denotes The Weight Of The Edge. The shortest pair edge disjoint paths in the new graph corresponds to the required solution in the original graph. the path itself, not just its length) between the source vertex given in from, to the target vertices given in to. A subgraph is a subset of a graph’s edges (with associated vertices) that form a graph. Length of a path is the sum of the weights of its edges. Let G = (V, E) be a directed graph and let v and w be two vertices in G. We present an improved algorithm for maintaining all-pairs (1 + ε) approximate shortest paths under deletions and weight-increases. weighted › cyclic vs. num_vertices() 11 for i in range(num_vertices): 12 result. Finding paths. The Bellman-Ford algorithm is a single-source shortest path algorithm. Topological Sort: Arranges the nodes in a directed, acyclic graph in a special order based on incoming edges. We will see how simple algorithms like depth-ﬁrst-search can be used in clever ways (for a problem known as topological sorting) and will see how Dynamic Programming can be used to solve problems of ﬁnding shortest paths. Asked in Graphs , C Programming. Dijkstra’s Algorithm for Finding the Shortest Path Through a Weighted Graph E. Simple path is a path with no repeated vertices. This algorithm aims to find the shortest-path in a directed or undirected graph with non-negative edge weights. BFS runs in O(E+V) time where E is the number of edges and V is number of vertices in the graph. Those times are the weights of those paths. A path in a graph is a sequence of adjacent vertices. We revisit a classical graph-theoretic problem, the single-source shortest-path (SSSP) problem, in weighted unit-disk graphs. Floyd-Warshall algorithm is an algorithm for finding shortest paths in a weighted graph with positive or negative edge weights (but with no negative cycles). Hi I have already posted a similar question but there was a misunderstanding of the problem by my side so here I post it again. Mark Dolan CIS 2166 10. weighted › cyclic vs. The N x N matrix of predecessors, which can be used to reconstruct the shortest paths. MINIMUM-WEIGHT SPANNING TREE 49 4. Year 2001 Predicted year 2005 Graph 2: Evolution of flu epidemic behavior. CS 312 Lecture 26 Finding Shortest Paths Finding Shortest Paths. C++ Program to Generate a Random UnDirected Graph for a Given Number of Edges; Shortest Path in a Directed Acyclic Graph. The weight of the path P is the sum of the weights of all the _____ on the path P. This problem could be solved easily using (BFS) if all edge weights were ($$1$$), but here weights can take any value. The length of a path is the sum of the lengths of all component edges. The weighted inverse distance is the total number of shortest paths from node \(s G From the Betweenness Centrality in Street Networks to Structural Invariants in Random Planar Graphs. In order to solve the load-balancing problem for coarse-grained parallelization, the relationship between the computing time of a single-source shortest-path length of node and the features of node is studied. So, the shortest path would be of length 1 and BFS would correctly find this for us. The total length of a path is the sum of the lengths of its component edges. In the notes below I am going to describe the Dijkstra algorithm, which is a widely-used algorithm for finding shortest paths in weighted, directed graphs. 2 commits 1 branch. In this post, I explain the single-source shortest paths problems out of the shortest paths problems, in which we need to find all the paths from one starting vertex to all other vertices. 7 (Single-Source Shortest Paths). The Degree of a vertex is the number of edges incident on it. Single-Source Shortest Path on Weighted Graphs. On weighted graphs Weighted Shortest Paths The shortest path from a vertex u to a vertex v in a graph is a path w1 = u, w2,…,wn= v, where the sum: Weight(w1,w2)+…+Weight(wn-1,wn) attains its minimal value among all paths that start at u and end at v The length of a path of n vertices is n-1 (the number of edges) If a graph is connected, and the weights are all non-negative, shortest paths exist for any pair of vertices Similarly for strongly connected digraphs with non-negative weights. Write an algorithm to print all possible paths between source and destination. Floyd-Warshall algorithm is used to find all pair shortest path problem from a given weighted graph. In real-life scenarios, the arc weighs in a shortest path of a network/graph have the several parameters which are very hard to define exactly (i. $\begingroup$ Shortest Path on an Undirected Graph? might be interesting. Weighted graphs and path length Weighted graphs A weighted graph is a graph whose edges have weights. Cover Photo by Thor Alvis on Unsplash. The path graph with n vertices is denoted by P n. In this tutorial, we will present a general explanation of both algorithms. Implementation of Dijkstra's algorithm in C++ which finds the shortest path from a start node to every other node in a weighted graph. There is a simple tweak to get from DFS to an algorithm that will find the shortest paths on an unweighted graph. the path itself, not just its length) between the source vertex given in from, to the target vertices given in to. (Consider what this means in terms of the graph shown above right. Furthermore, if we perform relaxation on the set of edges once, then we will at least have determined all the one-edged shortest paths; if we traverse the set of edges twice, we will have solved at least all the two-edged shortest paths; ergo, after the V-1 iteration. However, most communication networks are dynamic, i. shortest path functions use it as the cost of the path; community finding methods use it as the strength of the relationship between two vertices, etc. More Algorithms for All-Pairs Shortest Paths in Weighted Graphs Timothy M. Shortest paths in an edge-weighted digraph 4->5 0. Graphs can be weighted (edges carry values) and directional (edges have direction). The Degree of a vertex is the number of edges incident on it. In the weighted matching [G85b, GT89, GT91] and maxi-mum ﬂow problems [GR98], for instance, the best algorithms for real- and integer-weighted graphs have running times diﬀering by a polynomial factor. Implementation of Dijkstra's algorithm in C++ which finds the shortest path from a start node to every other node in a weighted graph. $\begingroup$ Shortest Path on an Undirected Graph? might be interesting. Simple path is a path with no repeated vertices. Is there a cycle in the graph? Euler tour. So, the shortest path would be of length 1 and BFS would correctly find this for us. First, the paths should be shortest, then there might be more than one such shortest paths whose length are the same. A path in a graph is a sequence of adjacent vertices. Mark Dolan CIS 2166 10. goldberg_radzik (G, source[, weight]) Compute shortest path lengths and predecessors on shortest paths in weighted graphs. Increasingly, there is interest in using asymmetric structure of data derived from Markov chains and directed graphs, but few metrics are specifically adapted to this task. We may also want to associate some cost or weight to the traversal of an edge. It is a real time graph algorithm, and can be used as part of the normal user flow in a web or mobile application. You may print the results of this algorithm to the screen or to a log file. shortest_paths calculates a single shortest path (i. However, the robot had a means for estimating Euclidean distances in his world. PY - 2007/10/30. More Algorithms for All-Pairs Shortest Paths in Weighted Graphs Timothy M. Based on Data Structures, Algorithms & Software Principles in CT. Finding shortest paths in weighted graphs In the past two weeks, you've developed a strong understanding of how to design classes to represent a graph and how to use a graph to represent a map. We ˙rst propose an exact (and deterministic) al-. In any graph G, the shortest path from a source vertex to a destination vertex can be calculated using Dijkstra Algorithm. , capacity, cost, demand, traffic frequency, time, etc. A subgraph is a subset of a graph’s edges (with associated vertices) that form a graph. Path Graphs. dijkstra_path¶ dijkstra_path (G, source, target, weight='weight') [source] ¶. This article presents a Java implementation of this algorithm. The previous state of the art for this problem was total update time O ̃(n 2 √m/ε) for directed, unweighted graphs [2], and Õ(mn=ε) for undirected, unweighted graphs [12]. Weighted Graphs Data Structures & Algorithms 2 [email protected] ©2000-2009 McQuain Shortest Paths (SSAD) Given a weighted graph, and a designated node S, we would like to find a path of least total weight from S to each of the other vertices in the graph. Chapter 4 Algorithms in edge-weighted graphs Recall that anedge-weighted graphis a pair(G,w)whereG=(V,E)is a graph andw:E →IR number of edges in a shortest path. , capacity, cost, demand, traffic frequency, time, etc. So, for this purpose we use the retroactive priority queue which allows to perform opeartions at any point of time. The Degree of a vertex is the number of edges incident on it. Shortest path algorithms are a family of algorithms designed to solve the shortest path problem. BFS always visits nodes in increasing order of their distance from the source. These values become important when calculating the. weighted graphs: ﬁnd a path from a given source to a given target such that the consecutive weights on the path are nondecreasing and the last weight on the path is minimized. A single execution of the algorithm will. Analyze your algorithm. ) B C A E D F 0 7 2 3 5 8 8 4 7 1 2 5 2. , its number of edges. Finding the Shortest Path in Weighted Directed Acyclic Graph For the graph above, starting with vertex 1, what're the shortest paths(the path which edges weight summation is minimal) to vertex 2. The Classical Dijkstra’s algorithm [16] solves the single-source shortest path problems in a simple graph. Shortest Distance in a graph with two different weights : Given a weighted undirected graph having A nodes, a source node C and destination node D. If G is a weighted graph, the length/weight of a path is the sum of the weights of the edges that compose the path. Dimitrios Skrepetos, PhD candidate David R. Check the manual pages of the functions working with weighted graphs for details. Another source vertex is also provided. A cycle is a path where the first and last vertices are the same. A subgraph is a subset of a graph’s edges (with associated vertices) that form a graph. Print the number of shortest paths from a given vertex to each of the vertices. However, the resulting algorithm is no longer called DFS. If There Is An Edge Between I And 1, Then G[16] > 0, Otherwise G[i,j]=-1. The weight of the path P is the sum of the weights of all the _____ on the path P. We will see how simple algorithms like depth-ﬁrst-search can be used in clever ways (for a problem known as topological sorting) and will see how Dynamic Programming can be used to solve problems of ﬁnding shortest paths. Dijkstra's Algorithm. ; It uses a priority based set or a queue to select a node / vertex nearest to the source that has not been edge relaxed. But how should we evaluate a path in a weighted graph? Usually, a path is assessed by the sum of weights of its edges and, based on this assumption, many authors proposed their. A graph is a series of nodes connected by edges. (2018) A Faster Distributed Single-Source Shortest Paths Algorithm. One algorithm for finding the shortest path from a starting node to a target node in a weighted graph is Dijkstra's algorithm. Count the number of updates At each step we compute the shortest path through a subset of vertices. When There Is An Edge Between I And J, Then G[i][j] Denotes The Weight Of The Edge. In this tutorial, we will present a general explanation of both algorithms. Single source shortest path for undirected graph is basically the breadth first traversal of the graph. Assignment: Given any connected, weighted graph G, use Dijkstras algorithm to compute the shortest (or smallest weight) path from any vertex a to any other vertex b in the graph G. If There Is An Edge Between I And 1, Then G[16] > 0, Otherwise G[i,j]=-1. G∗ contains threeshortcuts: v8,v9, v9,v7,and v9,v10. One problem might be the shortest path in a given undirected, weighted graph. GUVEW ,, eE , where (non-negative real number) is a weight function, by which each edge is associ-ated with a weight The weight of a matching M is. 1 def shortest_path_cycle(graph, s): 2 '''Single source shortest paths using DP on a graph with cycles but no 3 negative cycles. num_vertices() 11 for i in range(num_vertices): 12 result. We consider a specific infinite graph here, namely the honeycomb grid. Fine the shortest weighted path from a vertex, s, to every other vertex in the graph. We introduce a metric on the. The N x N matrix of predecessors, which can be used to reconstruct the shortest paths. There are already comprehensive network analysis packages in R, notably igraph and its tidyverse-compatible interface tidygraph. None of these algorithms for the WRP permit one to bound the number of links/turns in the produced path.
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# How to prove $n!>(\frac{n}{e})^{n}$

Prove that $n!>\left(\dfrac{n}{e}\right)^{n}$.

I used induction principle but cannot solve it for the $(m+1)$-th term after taking the $m$th term to be true.

Here the key is to use the appropriate definition of $e^x$, namely:

$$e^x = \sum_{k=0}^{\infty}\frac{1}{k!}x^k$$

Plugging in $x = n$ we get

$$e^n = \sum_{k=0}^{\infty} \frac{1}{k!}n^k$$

and hence, breaking this sum up a little we get our inequality: $$n! e^n = n^n + \sum_{k\ne n} \frac{n!}{k!}n^k > n^n$$

Hint: Show that $$\ln(n!)=\sum_{k=1}^n \ln k >\int_1^n\ln x\, dx.$$

• That's a clever approach! Nice one! Aug 25 '13 at 18:02

Inductively, if $n!>\frac{n^n}{e^n}$ and you multiply both sides by $n+1$, then you have that $(n+1)!>(n+1)\frac{n^n}{e^n}$, so it suffices to prove that $(n+1)\frac{n^n}{e^n}>\frac{(n+1)^{n+1}}{e^{n+1}}$. Can you continue from here?

• +1 This requires the least amount of machinery of the presented solutions, I think. Aug 25 '13 at 17:11

Hint: write out the series for $e^n$ and pick out a relevant term amongst the positive terms which make up the sum.

• You posted a hint and Deven independently posted a solution using the same idea => 10 vote difference. Isn't that always so? +1 to both of you from me has been there from the beginning, of course. Aug 25 '13 at 18:10

I had originally written this up for another question but it seems fitting here as well. Maybe this can help someone.

Depending on how you introduced $e$, you might be able to use the fact that there are two sequences $(a_n)_{n \in \mathbb{N}}$, $(b_n)_{n \in \mathbb{N}}$ with

\begin{align} a_n ~~~&:=~~~ \left ( 1 + \frac{1}{n} \right ) ^n \\ ~ \\ b_n ~~~&:=~~~ \left ( 1 - \frac{1}{n} \right ) ^{-n} \end{align}

and

$$\underset{n \rightarrow \infty}{\lim} a_n ~~~=~~~ \underset{n \rightarrow \infty}{\lim} b_n ~~~=~~~ e \\ ~ \\$$

While both sequences converge to the same limit, $a_n$ approaches from the bottom and $b_n$ approaches from the top:

import numpy as np
import matplotlib.pyplot as plt
from matplotlib import rcParams
rcParams.update({'figure.autolayout': True})

pts = np.arange(0, 20, 1)
a_n = lambda n: (1+1/n)**n
b_n = lambda n: (1-1/n)**(-n)

plt.errorbar(x = pts, xerr = None, y = a_n(pts), yerr = None, fmt = "bx", markersize = "5", markeredgewidth = "2", label = "$a_n$")
plt.errorbar(x = pts, xerr = None, y = b_n(pts), yerr = None, fmt = "rx", markersize = "5", markeredgewidth = "2", label = "$b_n$")
plt.plot(pts, [np.exp(1)]*len(pts), color = "black", linewidth = 2, label = "$e$")
plt.xlim(1.5, 14.5)
plt.ylim(2.0, 3.5)
plt.legend(loc = "best")
plt.setp(plt.gca().get_legend().get_texts(), fontsize = "22")
plt.show()

So we're going to use the following inequality:

$$\forall n \in \mathbb{N} ~ : ~~~~~ \left ( 1 + \frac{1}{n} \right ) ^n ~~~~<~~~~ e ~~~~<~~~~ \left ( 1 - \frac{1}{n} \right ) ^{-n} \tag*{\circledast} \\ ~ \\$$

Thesis

$$\forall n \in \mathbb{N}, ~ n \geq 2 ~ : ~~~~~ e \cdot \left ( \frac{n}{e} \right )^n ~~~~<~~~~ n! ~~~~<~~~~ n \cdot e \cdot \left ( \frac{n}{e} \right )^n \\ ~ \\$$

Proof By Induction

Base Case

We begin with $n = 2$ and get

\begin{align} & ~ && e \cdot \left ( \frac{2}{e} \right )^2 ~~~~&&<~~~~ 2! ~~~~&&<~~~~ 2 \cdot e \cdot \left ( \frac{2}{e} \right )^2 \\ ~ \\ & \Leftrightarrow && e \cdot \frac{4}{e^2} ~~~~&&<~~~~ 1 \cdot 2 ~~~~&&<~~~~ 2 \cdot e \cdot \frac{4}{e^2} \\ ~ \\ & \Leftrightarrow && \frac{4}{e} ~~~~&&<~~~~ 2 ~~~~&&<~~~~ \frac{8}{e} \\ ~ \\ &\Leftrightarrow && 2 ~~~~&&<~~~~ e ~~~~&&<~~~~ 4 ~~~~ \\ \end{align}

Which is a true statement.

Inductive Hypothesis

Therefore the statement holds for some $n$. $\tag*{$\text{I.H.}$}$

Inductive Step

\begin{align} & ~ && e \cdot \left ( \frac{n+1}{e} \right )^{n+1} \\ ~ \\ & = && (n+1) \cdot \frac{1}{e} \cdot e \cdot \left ( \frac{n+1}{e} \right )^n\\ ~ \\ & = && (n+1) \cdot \left ( \frac{n}{e} \right )^n \cdot \left ( \frac{n+1}{n} \right )^n\\ ~ \\ & = && (n+1) \cdot \left ( \frac{n}{e} \right )^n \cdot \left ( 1 + \frac{1}{n} \right )^n\\ ~ \\ & \overset{\circledast}{<} && (n+1) \cdot \left ( \frac{n}{e} \right )^n \cdot e\\ ~ \\ & \overset{\text{I.H.}}{<} && (n+1) \cdot n!\\ ~ \\ & = && (n+1)!\\ ~ \\ & = && (n+1) \cdot n!\\ ~ \\ & \overset{\text{I.H.}}{<} && (n+1) \cdot n \cdot e \cdot \left ( \frac{n}{e} \right )^n\\ ~ \\ & = && (n+1) \cdot e \cdot \left ( \frac{n}{e} \right )^{n+1} \cdot e \\ ~ \\ & = && (n+1) \cdot e \cdot \left ( \frac{n+1}{e} \right )^{n+1} \cdot \left ( \frac{n}{n+1} \right )^{n+1} \cdot e \\ ~ \\ & = && (n+1) \cdot e \cdot \left ( \frac{n+1}{e} \right )^{n+1} \cdot \left ( 1 - \frac{1}{n+1} \right )^{n+1} \cdot e \\ ~ \\ & \overset{\circledast}{<} && (n+1) \cdot e \cdot \left ( \frac{n+1}{e} \right )^{n+1} \cdot \left ( 1 - \frac{1}{n+1} \right )^{n+1} \cdot \left ( 1 - \frac{1}{n+1} \right )^{-(n+1)} \\ ~ \\ & = && (n+1) \cdot e \cdot \left ( \frac{n+1}{e} \right )^{n+1} \\ ~ \\ \end{align}

Conclusion

Therefore the statement holds $\forall n \in \mathbb{N}, ~ n \geq 2$. $$\tag*{\square}$$


# Probability of drawing cards on specific draw counts

There is a deck of $30$ cards, each card labeled a number from $1$ to $15$, with exactly $2$ copies of a card for each number. You draw $8$ cards. What is the probability that you draw the number '$1$' card by the $5$th draw (on the $5$th draw or before that), AND also drawing the number '$2$' card on or before the $8$th draw?

I know how to compute the probability of drawing both the cards on or before the $5$th draw:

$$\frac{\binom{2}{1}\cdot \binom{2}{1} \cdot \binom{26}{3}}{\binom{30}{5}}$$

Since there's $2$ ways to choose from each of the '$1$' and '$2$' cards, and then there's $26$ cards left after those $4$ cards so the other $3$ cards can be any of those $26$, and the total number of combinations you can draw $5$ cards from $30$.

But we want to expand this search to $8$ draws, and also at the same time want to have assumed that we have already drawn the '$1$' card on or before the $5$th draw (if we don't get the '$2$' card by the $5$th draw. How can I combine these ideas? Thanks

• number 2 before 8-th draw... Then the 8th draw is irrelevant? Don't you mean (again) "on or before"? – drhab Sep 3 '17 at 9:07
• @user152294 Just to check my try, have you the result of this exercise? – Robert Z Sep 3 '17 at 9:46
• @drhab Yes, on or before – user152294 Sep 3 '17 at 17:29
• @RobertZ No, I don't have the solution unfortunately – user152294 Sep 3 '17 at 17:30
• @user152294 "before 8-th draw" means "on or before 8-th draw"? In case I have to modify my solution. P.S. Where does his exercise come from? – Robert Z Sep 3 '17 at 17:33

I think it is more convenient to evaluate the probability of the complementary event: draw NO number '1' card by the $5$th draw, OR draw NO number '2' card. Here we consider the 2 copies of a card with the same number distinguishable (for example assume that one is red and the other is blue). Let $n^{\underline{k}}:=n(n-1)\cdots(n-k+1)$.

1) If we draw NO number '1' card by the 5th draw then we can have zero, one or two '1's in the $6$th, $7$th or $8$th draw $$p_1=\frac{1}{30^{\underline{8}}}\left(28^{\underline{8}}+(3+3)\cdot 28^{\underline{7}}+3\cdot 2\cdot 28^{\underline{6}}\right)$$

2) If we draw NO number '2' card then $$p_2=\frac{28^{\underline{8}}}{30^{\underline{8}}}$$

3) If we draw NO number '1' card by the $5$th draw AND NO number '2' card then, similarly to case 1), $$p_3=\frac{1}{30^{\underline{8}}}\left(26^{\underline{8}}+(3+3)\cdot 26^{\underline{7}}+3\cdot 2\cdot 26^{\underline{6}}\right)$$

Hence, the desired probability is $$p=1-(p_1+p_2-p_3)=211/1566\approx 0.134738.$$

• Is it also possible to just do these two cases? 1) Draw a '1' and '2' before the 5th draw, or 2) Draw a '1' before the 5th draw AND draw a '2' before the 8th draw? I'm not sure how to express 2) but would it be more cumbersome than the complementary events? – user152294 Sep 3 '17 at 17:34
• @user152294 Yes you can consider two cases, but I think that with 3 cases is simpler. – Robert Z Sep 3 '17 at 17:36
• How come in this solution, we don't have to use binomial coefficients? – user152294 Sep 3 '17 at 17:43
• @user152294 Let me know if the official solution is the one that I have found. – Robert Z Sep 3 '17 at 17:44
• @user152294 $3$ is the number of ways to place the red $1$ (position 6,7, 8), $3$ is the number of ways to place the blue $1$ (position 6,7, 8) and $3\cdot 2$ is the number of ways to place the red $1$ and blue $1$ (positions (6,7), (6,8), (7,8), (7,6), (8,6), (8,7)). $28^{\underline{8}}$, $28^{\underline{7}}$ and $28^{\underline{6}}$ are the number of ways to fill the remaining positions (cards different from 1). – Robert Z Sep 3 '17 at 18:21


# What does $f|_A$ mean? [duplicate]

If $f$ a is a function and $A$ is a set, what could the notation

$$f|_A$$

mean? Is it perhaps "restricted to set $A$"?

• Yes, if $f$ is a function with domain $D$, then for a set $A \subseteq D$, $f|_A$ is used for the restriction of $f$ to $A$. Feb 2, 2016 at 15:43
• That's what I would assume. You might have two functions $f$ and $g$ for which $f \neq g$ but $f|_A = g|_A$ for some $A$. Feb 2, 2016 at 15:43

It means that I am constricting the domain of the function $f$. If $f:X\to Y$, then $g=f|_A$ means that $g:A\to Y$ where $A\subseteq X$.

Intuitively speaking, a function $f$ is constituted of three ingredients:

• a domain;
• a codomain;
• a rule (that, for each element in the domain, assigns a unique element in the codomain).

If we change any of these three ingredients, we obtain a different function. In particular, if we change the domain by a subset $A$ of the original domain (keeping the codomain and the rule), we get a new function which is represented by $f|_A$.

In other words: given a function $f:X\to Y$ and a set $A\subset X$, the notation $f|_A$ denotes the function $g:A\to Y$ given by $$g(x)=f(x),\quad \forall \ x\in A.$$

This is the usual meaning but, maybe, there are different meanings in other contexts.

That's correct.

Suppose we have a function $$f : Y \leftarrow X,$$ and a subset $A$ of $X$.

Approach 0. Then $f \restriction_A$ is defined as the unique function $Y \leftarrow A$ that agrees with $f$ on $A$. That is: $$\mathop{\forall}_{a \in A} ((f \restriction_A)(a) = f(a))$$

However, there's a cleaner way of formalizing this.

Approach 1. Write $$\mathrm{incl}_A : X \leftarrow A$$ for the inclusion of $A$ into $X$. Then we can form the composite $$f \circ \mathrm{incl}_A : Y \leftarrow A.$$ Write $f\restriction_A$ as a shorthand for this composite.

The nice thing about Approach 1 is that it makes proving the basic properties of the restriction operator trivial. In particular:

Claim. $(g \circ f)\restriction_A = g \circ (f \restriction_A)$

Proof.

$$(g \circ f)\restriction_A = (g \circ f) \circ \mathrm{incl}_A = g \circ (f \circ \mathrm{incl}_A) = g \circ (f \restriction_A)$$

Approach 1 is especially appealing from a category-theory perspective. In that context:

• $X$ is an object of some category
• a subobject of $X$ is by definition an object $\underline{A}$ together with a monomorphism $\mathrm{incl}_A : X \leftarrow \underline{A}$.
• a partial morphism $Y \leftarrow X$ consists of a subobject $A$ of $X$ together with a morphism $Y \leftarrow \underline{A}$.

Hence, if we're given a morphism $f : Y \leftarrow X$ and a subobject $A$ of $X$, then we get a partial morphism $f \restriction_A : Y \leftarrow X$ by forming the obvious composite.

The notation $f|_A$ is probably best understood via a meaningful example. Before giving one (I hope it will be useful, anyway), it would probably be good to consult two decent references:

2) Abstract Algebra by Dummit and Foote (p. 3, 3rd Ed.).

The relevant portion from the Wiki blurb:

Let $f\colon E\to F$ be a function from a set $E$ to a set $F$, so that the domain of $f$ is in $E$ (i.e., $\operatorname{dom}f\subseteq E$). If $A\subseteq E$, then the restriction of $f$ to $A$ is the function $f|_A\colon A\to F$.

Informally, the restriction of $f$ to $A$ is the same function as $f$, but is only defined on $A\cap\operatorname{dom} f$.

Wiki's "informal" remark is the key part in my opinion. The following excerpt from Dummit and Foote's Abstract Algebra may be slightly more abstract, but I think a meaningful example will clear everything up.

If $A\subseteq B$, and $f\colon B\to C$, we denote the restriction of $f$ to $A$ by $f|_A$. When the domain we are considering is understood we shall occasionally denote $f|_A$ again simply as $f$ even though these are formally different functions (their domains are different).

If $A\subseteq B$ and $g\colon A\to C$ and there is a function $f\colon B\to C$ such that $f|_A=g$, we shall say $f$ is an extension of $g$ to $B$ (such a map $f$ need not exist nor be unique).

Example: Let $g\colon\mathbb{Z}^+\to\{1\}$ be defined by $g(x)=1$ and let $f\colon\mathbb{Z}\setminus\{0\}\to\{1\}$ be defined by $f(x)=\dfrac{|x|}{x}$. Using the notation from the second paragraph above, we have $g\colon A\to C$ and $f\colon B\to C$, where

• $A = \mathbb{Z^+}$
• $B=\mathbb{Z}\setminus\{0\}$
• $C=\{1\}$

and, clearly, $A\subseteq B$. Thus, we have the following: \begin{align} f|_A &\equiv f\colon\mathbb{Z^+}\to\{1\}\tag{by definition}\\[0.5em] &= \frac{|x|}{x}\tag{by definition}\\[0.5em] &= \frac{x}{x}\tag{if $x\in\mathbb{Z^+}$, then $|x|=x$ }\\[0.5em] &= 1\tag{simplify}\\[0.5em] &\equiv g\colon\mathbb{Z^+}\to\{1\}\tag{by definition}\\[0.5em] &= g. \end{align} Apart from some slight notational abuse, perhaps, the above example shows that $f$ is an extension of $g$ to $B$ since $f|_A=g$.


Mean and Variance of "Piecewise" Normal Distribution

Note - I put piecewise in quotes because I don't think it's the right term to use (I can't figure out what to call it).

I am building a program to model the load that a user places on a server. The load a user produces follows a normal distribution. Depending on which application the user is using, however, the mean and variance of that normal distribution will be different. What I am trying to do is calculate the overall mean and variance for a user's activity given the proportions of time they use each application.

For example, Application A follows $\mathcal{N}(100, 50)$ and Application B follows $\mathcal{N}(500, 20)$. If a user uses A 50% of the time and B the other 50%, what is the mean and variance of the data that the user would produce during a day?

I'm able to simulate this by selecting a number from a uniform distribution between 0 and 1 and then generating a value from the appropriate distribution. Something like this:

$f(x) = \begin{cases} \mathcal{N}(100, 50), &0 \le x \lt 0.5\\ \mathcal{N}(500, 20), &0.5 \le x \lt 1\\ \end{cases}$

When I simulate a large number of these values and measure the results, it looks like the mean is just

$\sum\limits_{i=1}^n\mu_ip$

where $p$ is the percentage of the day a user is using each application.

I can't figure out what pattern the variance follows or what the formula might be to determine it without measuring a bunch of simulated values (When I simulate the above example, the variance looks to be something close to 41500).

I'd appreciate confirmation that how I'm calculating the combined mean is correct and some help in figuring out how to determine the variance of the overall distribution.

Let the two normal random variables be $X$ and $Y$, where $X$ is chosen with probability $p$, and $Y$ is chosen with probability $q=1-p$.

If $W$ is the resulting random variable, then $\Pr(W\le w)=p\Pr(X\le w)+q\Pr(Y\le w)$.

Differentiate. We get $f_W(w)=pf_X(w)+qf_Y(w)$.

The mean of $W$ is $\int_{-\infty}^\infty wf_W(w)$. Calculate. We get $$\int_{-\infty}^\infty w(pf_X(w)+qf_Y(w))\,dw.$$ This is $pE(X)+qE(Y)$, confirming your observation.

For the variance, we want $E(W^2)-(E(W))^2$. For $E(W^2)$, we calculate $\int_{-\infty}^{\infty} w^2(pf_X(w)+qf_Y(w))\,dw$. This is $pE(X^2)+qE(Y^2)$.

But $pE(X^2)= p(\text{Var}(X)+(E(X))^2)$ and $qE(Y^2)= q(\text{Var}(Y)+(E(Y))^2)$

Putting things together we get $$\text{Var}(W)=p\text{Var}(X)+q\text{Var}(Y)+ p(E(X))^2+q(E(Y))^2- (pE(X)+qE(Y))^2.$$

Remark: For a longer discussion, please look for Mixture Distributions.

• So, to put it more generally: $\text{Var}(W) = \sum\limits_{i=1}^np_i(\text{Var}(X_i) + E(X_i)^2) - (\sum\limits_{i=1}^np_iE(X_i))^2$ Aug 5, 2014 at 17:57
• Yes, the same argument works for $X_i$ with probability $p_i$. One could generalize further, to sums to $\infty$, also to "continuous" mixtures. Mixtures happen a lot in Statistics, since populations can often be stratified. Aug 5, 2014 at 18:15
• Perfect. Thank you so much for your help! I already ran some simulations with a few different combinations and it works beautifully. Aug 5, 2014 at 18:47
• You are welcome. Fast work. Aug 5, 2014 at 19:49

What you are wanting are the distributional quantities of the mixture distribution. There are some convenient formulas to do this: let $T$ be the unconditional load a user places on the system, and let $X$ be a Bernoulli random variable that indicates whether a user is on system $A$ or $B$. So $A = (T \mid X = 0) \sim \mathrm{Normal}(\mu_A = 100, \sigma_A^2 = 50)$, and $B = (T \mid X = 1) \sim \mathrm{Normal}(\mu_B = 500, \sigma_B^2 = 20)$. That is to say, \begin{align*} \mathrm{E}[T \mid X = 0] &= \mu_A, \\ \mathrm{E}[T \mid X = 1] &= \mu_B, \\ \mathrm{Var}[T \mid X = 0] &= \sigma_A^2, \\ \mathrm{Var}[T \mid X = 1] &= \sigma_B^2. \end{align*}

Then by the law of total expectation, \begin{align*} \mathrm{E}[T] &= \mathrm{E}[\mathrm{E}[T \mid X]] \\ &= \mathrm{E}[T \mid X = 0]\Pr[X = 0] + \mathrm{E}[T \mid X = 1]\Pr[X = 1] \\ &= \mu_A (1-p) + \mu_B p,\end{align*} where $p$ is the probability that a user is on system $B$. The variance is calculated by \begin{align*} \mathrm{Var}[T] &= \mathrm{E}[\mathrm{Var}[T \mid X]] + \mathrm{Var}[\mathrm{E}[T \mid X]] \\ &= \mathrm{Var}[T \mid X = 0]\Pr[X = 0] + \mathrm{Var}[T \mid X = 1]\Pr[X = 1] + \mathrm{Var}[\mathrm{E}[T \mid X]] \\ &= \sigma_A^2 (1-p) \sigma_B^2 p + \mathrm{Var}[\mathrm{E}[T \mid X]]. \end{align*} The last term requires a little subtlety to understand. The variable $\mathrm{E}[T \mid X]$ is a generalized Bernoulli, which takes on the value $\mu_A$ with probability $1-p$ and $\mu_B$ with probability $p$ (rather than $0$ and $1$). So we may write this as $$\mathrm{E}[T \mid X] = X(\mu_B - \mu_A) + \mu_A,$$ where $X \sim \mathrm{Bernoulli}(p)$. Therefore, $$\mathrm{Var}[\mathrm{E}[T \mid X]] = \mathrm{Var}[(\mu_B - \mu_A)X + \mu_A] = (\mu_B - \mu_A)^2 \mathrm{Var}[X] = (\mu_B - \mu_A)^2 p(1-p).$$ Hence $$\mathrm{Var}[T] = \sigma_A^2 (1-p) + \sigma_B^2 p + (\mu_B - \mu_A)^2 p(1-p).$$ The general case with $n$ systems $S_i \sim \mathrm{Normal}(\mu_i, \sigma_i^2)$, $i = 1, 2, \ldots, n$, where the user is on system $i$ with a probability of $p_i$, with $\sum_{i=1}^n p_i = 1$, involves a categorical distribution rather than a Bernoulli.

Recall that $V[X]=E[X^2]-E[X]^2$. Hence if you are given $E$ and $V$ of $X_1,X_2$ and combine them as described to a new random variable $Y$ by using a 0-1-random variable $Z$, i.e. picking $X_1$ if $Z=1$ and picking $X_2$ if $Z=0$, we find $$E[Y]=P(Z=1)\cdot E[Y|Z=1]+P(Z=0)\cdot E[Y|Z=0]=pE[X_1]+(1-p)E[X_2].$$ By the same argument we find $$E[Y^2] = pE[X_1^2]+(1-p)E[X_2^2]$$ Substituting $E[X_i^2]=V[X_i]+E[X_i]^2$, we obtain \begin{align}V[Y]&=E[Y^2]-E[Y]^2 \\&= p(V[X_1]+E[X_1]^2) + (1-p)(V[X_2]+E[X_2]^2)-(pE[X_1]+(1-p)E[X_2])^2\\ &=pV[X_1]+(1-p)V[X_2]+p(1-p)(E[X_1]^2+E[X_2]^2)-2p(1-p)E[X_1]E[X_2]\\ &=pV[X_1]+(1-p)V[X_2]+p(1-p)(E[X_1]-E[X_2])^2.\end{align}