DOCSLIB.ORG

The Causal Interpretations of Bayesian Hypergraphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Causal Interpretations of Bayesian Hypergraphs The Causal Interpretations of Bayesian Hypergraphs Zhiyu Wang1, Mohammad Ali Javidian2, Linyuan Lu1, Marco Valtorta2 1Department of Mathematics 2Department of Computer Science and Engineering University of South Carolina, Columbia, SC, 29208 Abstract Probabilistic Graphical Models (PGMs) enjoy a well-deserved popularity because they allow ex- We propose a causal interpretation of Bayesian plicit representation of structural constraints in the hypergraphs (Javidian et al. 2018), a probabilis- tic graphical model whose structure is a directed language of graphs and similar structures. From acyclic hypergraph, that extends the causal interpre- the perspective of efficient belief update, factoriza- tation of LWF chain graphs. We provide intervention tion of the joint probability distribution of random formulas and a graphical criterion for intervention variables corresponding to variables in the graph is in Bayesian hypergraphs that specializes to a new paramount, because it allows decomposition of the graphical criterion for intervention in LWF chain calculation of the evidence or of the posterior prob- graphs and sheds light on the causal interpretation ability (Lauritzen and Jensen 1997). The prolifera- of interaction as represented by undirected edges tion of different PGMs that allow factorizations of in LWF chain graphs or heads in Bayesian hyper- different kinds leads us to consider a more general graphs. graphical structure in this paper, namely directed acyclic hypergraphs. Since there are many more Introduction and Motivation hypergraphs than DAGs, undirected graphs, chain Probabilistic graphical models are graphs in which graphs, and, indeed, other graph-based networks, nodes represent random variables and edges rep- Bayesian hypergraphs can model much finer fac- resent conditional independence assumptions. They torizations and thus are more computationally effi- provide a compact way to represent the joint prob- cient. See (Javidian et al. 2018) for examples of how ability distributions of a set of random variables. Bayesian hypergraphs can model graphically func- In undirected graphical models, e.g., Markov net- tional dependencies that are hidden in probability ta- works (see (Darroch et al. 1980; Pearl 1988)), there bles when using BNs or CGs. We provide a causal is a simple rule for determining independence: two interpretation of Bayesian hypergraphs that extends set of nodes A and B are conditionally independent the causal interpretation of LWF chain graphs (Lau- given C if removing C separates A and B. On the ritzen and Richardson 2002) by giving correspond- other hand, directed graphical models, e.g. Bayesian ing formulas and a graphical criterion for interven- networks (see (Kiiveri, Speed, and Carlin 1984; tion, which operationalize the intuition that feed- Wermuth and Lauritzen 1983; Pearl 1988)), which back processes in LWF chain graphs and Bayesian consist of a directed acyclic graph (DAG) and a hypergraphs models turn into causal processes when corresponding set of conditional probability tables, variables in them are conditioned upon by inter- have a more complicated rule (d-separation) for de- vention. The addition of feedback processes and its termining independence. More complex graphical causal interpretation is a conceptual advance within models include various types of graphs with edges the three-level causal hierarchy (Pearl 2009). of several types (e.g., (Cox and Wermuth 1993; 1996; Richardson and Spirtes 2002; Pena˜ 2014)), in- Bayesian Hypergraphs: Definitions cluding chain graphs (Lauritzen and Wermuth 1989; Hypergraphs are generalizations of graphs such that Lauritzen 1996), for which different interpretations each edge is allowed to contain more than two ver- have emerged (Andersson, Madigan, and Perlman tices. Formally, an (undirected) hypergraph is a pair 1996; Drton 2009). H = (V; E), where V = fv1; v2; ··· ; vng is the set of Copyright c 2019, Association for the Advancement of vertices (or nodes) and E = fh1; h2; ··· ; hmg is the Artificial Intelligence (www.aaai.org). All rights reserved. set of hyperedges where hi ⊆ V for all i 2 [m]. If jh j = k for every i 2 [m], then we say H is a k- chain components fτ : τ 2 Dg yields an unique nat- i S uniform (undirected) hypergraph. A directed hyper- ural partition of the vertex set V(H) = τ2D τ with edge or hyperarc h is an ordered pair, h = (X; Y), of the following properties: (possibly empty) subsets of V where X \ Y = ;; X Proposition 1. Let H be a DAH and fτ : τ 2 Dg is the called the tail of h while Y is the head of h. be its chain components. Let G be a graph obtained We write X = T(h) and Y = H(h). We say a directed from H by contracting each element of fτ : τ 2 Dg hyperedge h is fully directed if none of H(h) and into a single vertex and creating a directed edge T(h) are empty. A directed hypergraph is a hyper- from τi 2 V(G) to τ j 2 V(G) in G if and only if there graph such that all of the hyperedges are directed. A exists a hyperedge h 2 E(H) such that T(h) \ τi , ; (s; t)-uniform directed hypergraph is a directed hy- and H(h) \ τ j , ;. Then G is a DAG. pergraph such that the tail and head of every directed edge have size s and t respectively. For example, any Proof. See (Javidian et al. 2018). DAG is a (1; 1)-uniform hypergraph (but not vice Note that the DAG obtained in Proposition 1 is versa). An undirected graph is a (0; 2)-uniform hy- unique and given a DAH H we call such G the pergraph. Given a hypergraph H, we use V(H) and canonical DAG of H. E(H) to denote the the vertex set and edge set of H respectively. Definition 1. A Bayesian hypergraph is a triple We say two vertices u and v are co-head (or co- (V; H; P) such that V is a set of random variables, tail) if there is a directed hyperedge h such that H is a DAH on the vertex set V and P is a multivari- fu; vg ⊂ H(h) ( or fu; vg ⊂ T(h) respectively). Given ate probability distribution on V such that the local another vertex u , v, we say u is a parent of v, de- Markov property holds with respect to the DAH H, noted by u ! v, if there is a directed hyperedge h i.e., for any vertex v 2 V(H), 2 2 such that u T(h) and v H(h). If u and v are co- v y nd(v)ncl(v) j bd(v): (1) head, then u is a neighbor of v. If u; v are neighbors, we denote them by u−v. Given v 2 V, we define par- For a Bayesian hypergraph H whose underlying ent (pa(v)), neighbor (nb(v)), boundary (bd(v)), an- DAH is a LWF DAH, we call H a LWF Bayesian cestor (an(v)), anterior (ant(v)), descendant (de(v)), hypergraph. and non-descendant (nd(v)) for hypergraphs exactly the same as for graphs (and therefore use the same Bayesian hypergraphs factorizations names). The same holds for the equivalent concepts The factorization of a probability measure P accord- for τ ⊆ V. Note that it is possible that some vertex u ing to a Bayesian hypergraph is similar to that of is both the parent and neighbor of v. a chain graph. Before we present the factorization A partially directed cycle in H is a sequence property, let us introduce some additional terminol- u fv1; v2;::: vkg satisfying that vi is either a neighbor ogy. Given a DAH H, we use H to denote the or a parent of vi+1 for all 1 ≤ i ≤ k and vi ! vi+1 for undirected hypergraph obtained from H by replac- some 1 ≤ i ≤ k. Here vk+1 ≡ v1. We say a directed ing each directed hyperedge h = (A; B) of H into an hypergraph H is acyclic if H contains no partially undirected hyperedge A [ B. Given a family of sets directed cycle. For ease of reference, we call a di- F , define a partial order (F ; ≤) on F such that for rected acyclic hypergraph a DAH or a Bayesian hy- two sets A; B 2 F , A ≤ B if and only if A ⊆ B. Let pergraph structure (as defined in Definition 1). Note M(F ) denote the set of maximal elements in F , i.e., that for any two vertices u; v in a directed acyclic no element in M(F ) contains another element as hypergraph H, u can not be both the parent and subset. When F is a set of directed hyperedges, we neighbor of v otherwise we would have a partially abuse the notation to denote M(F ) = M(F u). Let directed cycle. H be a directed acyclic hypergraph and fτ : τ 2 Dg Remark 1. DAHs are generalizations of undirected be its chain components. Assume that a probability graphs, DAGs and chain graphs. In particular an distribution P has a density f , with respect to some undirected graph can be viewed as a DAH in which product measure µ = ×α2V µα on X = ×α2V Xα. Now every hyperedge is of the form (;; fu; vg). A DAG we say a probability measure P factorizes according is a DAH in which every hyperedge is of the form to H if it has density f such that (fug; fvg). A chain graph is a DAH in which every (i) f factorizes as in the directed acyclic case: hyperedge is of the form (;; fu; vg) or (fug; fvg). Y f (x) = f (x j x ): (2) We define the chain components of H as the τ pa(τ) 2D equivalence classes under the equivalence relation τ ∗ where two vertices v1; vt are equivalent if there ex- (ii) For each τ 2 D, define Hτ to be the subhy- ists a sequence of distinct vertices v1; v2;:::; vt such pergraph of Hτ[pa(τ) containing all edges h in that vi and vi+1 are co-head for all i 2 [t − 1].
Load more

Recommended publications
  • Applications of DFS
    (b) acyclic (tree) iff a DFS yeilds no Back edges 2.A directed graph is acyclic iff a DFS yields no back edges, i.e., DAG (directed acyclic graph) , no back edges 3. Topological sort of a DAG { next 4. Connected components of a undirected graph (see Homework 6) 5. Strongly connected components of a drected graph (see Sec.22.5 of [CLRS,3rd ed.]) Applications of DFS 1. For a undirected graph, (a) a DFS produces only Tree and Back edges 1 / 7 2.A directed graph is acyclic iff a DFS yields no back edges, i.e., DAG (directed acyclic graph) , no back edges 3. Topological sort of a DAG { next 4. Connected components of a undirected graph (see Homework 6) 5. Strongly connected components of a drected graph (see Sec.22.5 of [CLRS,3rd ed.]) Applications of DFS 1. For a undirected graph, (a) a DFS produces only Tree and Back edges (b) acyclic (tree) iff a DFS yeilds no Back edges 1 / 7 3. Topological sort of a DAG { next 4. Connected components of a undirected graph (see Homework 6) 5. Strongly connected components of a drected graph (see Sec.22.5 of [CLRS,3rd ed.]) Applications of DFS 1. For a undirected graph, (a) a DFS produces only Tree and Back edges (b) acyclic (tree) iff a DFS yeilds no Back edges 2.A directed graph is acyclic iff a DFS yields no back edges, i.e., DAG (directed acyclic graph) , no back edges 1 / 7 4. Connected components of a undirected graph (see Homework 6) 5.
    [Show full text]
  • 9 the Graph Data Model
    CHAPTER 9 ✦ ✦ ✦ ✦ The Graph Data Model A graph is, in a sense, nothing more than a binary relation. However, it has a powerful visualization as a set of points (called nodes) connected by lines (called edges) or by arrows (called arcs). In this regard, the graph is a generalization of the tree data model that we studied in Chapter 5. Like trees, graphs come in several forms: directed/undirected, and labeled/unlabeled. Also like trees, graphs are useful in a wide spectrum of problems such as com- puting distances, finding circularities in relationships, and determining connectiv- ities. We have already seen graphs used to represent the structure of programs in Chapter 2. Graphs were used in Chapter 7 to represent binary relations and to illustrate certain properties of relations, like commutativity. We shall see graphs used to represent automata in Chapter 10 and to represent electronic circuits in Chapter 13. Several other important applications of graphs are discussed in this chapter. ✦ ✦ ✦ ✦ 9.1 What This Chapter Is About The main topics of this chapter are ✦ The definitions concerning directed and undirected graphs (Sections 9.2 and 9.10). ✦ The two principal data structures for representing graphs: adjacency lists and adjacency matrices (Section 9.3). ✦ An algorithm and data structure for finding the connected components of an undirected graph (Section 9.4). ✦ A technique for finding minimal spanning trees (Section 9.5). ✦ A useful technique for exploring graphs, called “depth-first search” (Section 9.6). 451 452 THE GRAPH DATA MODEL ✦ Applications of depth-first search to test whether a directed graph has a cycle, to find a topological order for acyclic graphs, and to determine whether there is a path from one node to another (Section 9.7).
    [Show full text]
  • A Low Complexity Topological Sorting Algorithm for Directed Acyclic Graph
    International Journal of Machine Learning and Computing, Vol. 4, No. 2, April 2014 A Low Complexity Topological Sorting Algorithm for Directed Acyclic Graph Renkun Liu then return error (graph has at Abstract—In a Directed Acyclic Graph (DAG), vertex A and least one cycle) vertex B are connected by a directed edge AB which shows that else return L (a topologically A comes before B in the ordering. In this way, we can find a sorted order) sorting algorithm totally different from Kahn or DFS algorithms, the directed edge already tells us the order of the Because we need to check every vertex and every edge for nodes, so that it can simply be sorted by re-ordering the nodes the “start nodes”, then sorting will check everything over according to the edges from a Direction Matrix. No vertex is again, so the complexity is O(E+V). specifically chosen, which makes the complexity to be O*E. An alternative algorithm for topological sorting is based on Then we can get an algorithm that has much lower complexity than Kahn and DFS. At last, the implement of the algorithm by Depth-First Search (DFS) [2]. For this algorithm, edges point matlab script will be shown in the appendix part. in the opposite direction as the previous algorithm. The algorithm loops through each node of the graph, in an Index Terms—DAG, algorithm, complexity, matlab. arbitrary order, initiating a depth-first search that terminates when it hits any node that has already been visited since the beginning of the topological sort: I.
    [Show full text]
  • CS302 Final Exam, December 5, 2016 - James S
    CS302 Final Exam, December 5, 2016 - James S. Plank Question 1 For each of the following algorithms/activities, tell me its running time with big-O notation. Use the answer sheet, and simply circle the correct running time. If n is unspecified, assume the following: If a vector or string is involved, assume that n is the number of elements. If a graph is involved, assume that n is the number of nodes. If the number of edges is not specified, then assume that the graph has O(n2) edges. A: Sorting a vector of uniformly distributed random numbers with bucket sort. B: In a graph with exactly one cycle, determining if a given node is on the cycle, or not on the cycle. C: Determining the connected components of an undirected graph. D: Sorting a vector of uniformly distributed random numbers with insertion sort. E: Finding a minimum spanning tree of a graph using Prim's algorithm. F: Sorting a vector of uniformly distributed random numbers with quicksort (average case). G: Calculating Fib(n) using dynamic programming. H: Performing a topological sort on a directed acyclic graph. I: Finding a minimum spanning tree of a graph using Kruskal's algorithm. J: Finding the minimum cut of a graph, after you have found the network flow. K: Finding the first augmenting path in the Edmonds Karp implementation of network flow. L: Processing the residual graph in the Ford Fulkerson algorithm, once you have found an augmenting path. Question 2 Please answer the following statements as true or false. A: Kruskal's algorithm requires a starting node.
    [Show full text]
  • Merkle Trees and Directed Acyclic Graphs (DAG) As We've Discussed, the Decentralized Web Depends on Linked Data Structures
    Merkle trees and Directed Acyclic Graphs (DAG) As we've discussed, the decentralized web depends on linked data structures. Let's explore what those look like. Merkle trees A Merkle tree (or simple "hash tree") is a data structure in which every node is hashed. +--------+ | | +---------+ root +---------+ | | | | | +----+---+ | | | | +----v-----+ +-----v----+ +-----v----+ | | | | | | | node A | | node B | | node C | | | | | | | +----------+ +-----+----+ +-----+----+ | | +-----v----+ +-----v----+ | | | | | node D | | node E +-------+ | | | | | +----------+ +-----+----+ | | | +-----v----+ +----v-----+ | | | | | node F | | node G | | | | | +----------+ +----------+ In a Merkle tree, nodes point to other nodes by their content addresses (hashes). (Remember, when we run data through a cryptographic hash, we get back a "hash" or "content address" that we can think of as a link, so a Merkle tree is a collection of linked nodes.) As previously discussed, all content addresses are unique to the data they represent. In the graph above, node E contains a reference to the hash for node F and node G. This means that the content address (hash) of node E is unique to a node containing those addresses. Getting lost? Let's imagine this as a set of directories, or file folders. If we run directory E through our hashing algorithm while it contains subdirectories F and G, the content-derived hash we get back will include references to those two directories. If we remove directory G, it's like Grace removing that whisker from her kitten photo. Directory E doesn't have the same contents anymore, so it gets a new hash. As the tree above is built, the final content address (hash) of the root node is unique to a tree that contains every node all the way down this tree.
    [Show full text]
  • Graph Algorithms G
    Bipartiteness Graph G = (V,E) is bipartite iff it can be partitioned into two sets of nodes A and B such that each edge has one end in A and the other Introduction to Algorithms end in B Alternatively: Graph Algorithms • Graph G = (V,E) is bipartite iff all its cycles have even length • Graph G = (V,E) is bipartite iff nodes can be coloured using two colours CSE 680 Question: given a graph G, how to test if the graph is bipartite? Prof. Roger Crawfis Note: graphs without cycles (trees) are bipartite bipartite: non bipartite Testing bipartiteness Toppgological Sort Method: use BFS search tree Recall: BFS is a rooted spanning tree. Algorithm: Want to “sort” or linearize a directed acyygp()clic graph (DAG). • Run BFS search and colour all nodes in odd layers red, others blue • Go through all edges in adjacency list and check if each of them has two different colours at its ends - if so then G is bipartite, otherwise it is not A B D We use the following alternative definitions in the analysis: • Graph G = (V,E) is bipartite iff all its cycles have even length, or • Graph G = (V, E) is bipartite iff it has no odd cycle C E A B C D E bipartit e non bipartite Toppgological Sort Example z PerformedonaPerformed on a DAG. A B D z Linear ordering of the vertices of G such that if 1/ (u, v) ∈ E, then u appears before v. Topological-Sort (G) 1. call DFS(G) to compute finishing times f [v] for all v ∈ V 2.
    [Show full text]
  • 6.042J Chapter 6: Directed Graphs
    “mcs-ftl” — 2010/9/8 — 0:40 — page 189 — #195 6 Directed Graphs 6.1 Definitions So far, we have been working with graphs with undirected edges. A directed edge is an edge where the endpoints are distinguished—one is the head and one is the tail. In particular, a directed edge is specified as an ordered pair of vertices u, v and is denoted by .u; v/ or u v. In this case, u is the tail of the edge and v is the ! head. For example, see Figure 6.1. A graph with directed edges is called a directed graph or digraph. Definition 6.1.1. A directed graph G .V; E/ consists of a nonempty set of D nodes V and a set of directed edges E. Each edge e of E is specified by an ordered pair of vertices u; v V . A directed graph is simple if it has no loops (that is, edges 2 of the form u u) and no multiple edges. ! Since we will focus on the case of simple directed graphs in this chapter, we will generally omit the word simple when referring to them. Note that such a graph can contain an edge u v as well as the edge v u since these are different edges ! ! (for example, they have a different tail). Directed graphs arise in applications where the relationship represented by an edge is 1-way or asymmetric. Examples include: a 1-way street, one person likes another but the feeling is not necessarily reciprocated, a communication channel such as a cable modem that has more capacity for downloading than uploading, one entity is larger than another, and one job needs to be completed before another job can begin.
    [Show full text]
  • Graph Algorithms
    Graph Algorithms Graphs and Graph Repre- sentations Graph Graph Algorithms Traversals COMP4128 Programming Challenges Directed Graphs and Cycles Strongly Connected Components Example: School of Computer Science and Engineering 2SAT UNSW Australia Minimum Spanning Trees Table of Contents 2 Graph Algorithms 1 Graphs and Graph Representations Graphs and Graph Repre- sentations 2 Graph Graph Traversals Traversals Directed Graphs and 3 Directed Graphs and Cycles Cycles Strongly Connected 4 Strongly Connected Components Components Example: 2SAT 5 Example: 2SAT Minimum Spanning Trees 6 Minimum Spanning Trees Graphs 3 Graph Algorithms Graphs and Graph Repre- sentations A graph is a collection of vertices and edges connecting Graph pairs of vertices. Traversals Directed Generally, graphs can be thought of as abstract Graphs and Cycles representations of objects and connections between those Strongly objects, e.g. intersections and roads, people and Connected Components friendships, variables and equations. Example: Many unexpected problems can be solved with graph 2SAT techniques. Minimum Spanning Trees Graphs 4 Graph Algorithms Graphs and Graph Repre- sentations Graph Many different types of graphs Traversals Directed graphs Directed Graphs and Acyclic graphs, including trees Cycles Weighted graphs Strongly Connected Flow graphs Components Other labels for the vertices and edges Example: 2SAT Minimum Spanning Trees Graph Representations 5 Graph Algorithms Graphs and Graph Repre- sentations Graph You’ll usually want to either use an adjacency list or an Traversals adjacency matrix to store your graph. Directed Graphs and An adjacency matrix is just a table (usually implemented Cycles as an array) where the jth entry in the ith row represents Strongly Connected the edge from i to j, or lack thereof.
    [Show full text]
  • The Shortest Path Problem with Edge Information Reuse Is NP-Complete
    The Shortest Path Problem with Edge Information Reuse is NP-Complete Jesper Larsson Tr¨aff TU Wien Faculty of Informatics, Institute of Information Systems Research Group Parallel Computing Favoritenstrasse 16/184-5, 1040 Vienna, Austria email: [email protected] 15.9.2015, Revised 6.6.2016 Abstract We show that the following variation of the single-source shortest path problem is NP- complete. Let a weighted, directed, acyclic graph G = (V,E,w) with source and sink vertices s and t be given. Let in addition a mapping f on E be given that associates information with the edges (e.g., a pointer), such that f(e)= f(e′) means that edges e and e′ carry the same information; for such edges it is required that w(e)= w(e′). The length of a simple st path U is the sum of the weights of the edges on U but edges with f(e)= f(e′) are counted only once. The problem is to determine a shortest such st path. We call this problem the edge information reuse shortest path problem. It is NP-complete by reduction from 3SAT. 1 The Edge Information Reuse Shortest Path Problem A weighted, directed, acyclic graph G = (V, E, w) with source and sink vertices s,t ∈ V is given. Edges represent some possible substructures and a path from s to t determines how substructures are put together to form a desired superstructure. Edge weights reflect the cost of the associated substructures (e.g., memory consumption). An ordinary shortest path determines an ordered, tree-like representation of the superstructure of least cost: a single root with children given by the edges of the path.
    [Show full text]
  • 5 Directed Acyclic Graphs
    5 Directed acyclic graphs (5.1) Introduction In many statistical studies we have prior knowledge about a temporal or causal ordering of the variables. In this chapter we will use directed graphs to incorporate such knowledge into a graphical model for the variables. Let XV = (Xv)v∈V be a vector of real-valued random variables with probability distribution P and density p. Then the density p can always be decomposed into a product of conditional densities, d Q p(x) = p(xd|x1, . , xd−1)p(x1, . , xd−1) = ... = p(xv|x1, . , xv−1). (5.1) v=1 Note that this can be achieved for any ordering of the variables. Now suppose that the conditional density of some variable Xv does not depend on all its predecessors, namely X1,...,Xv−1, but only on a subset XUv , that is, Xv is conditionally inde- pendent of its predecessors given XUv . Substituting p(xv|xUv ) for p(xv|x1, . , xv−1) in the product (5.1), we obtain d Q p(x) = p(xv|xUv ). (5.2) v=1 This recursive dependence structure can be represented by a directed graph G by drawing an arrow from each vertex in Uv to v. As an immediate consequence of the recursive factorization, the resulting graph is acyclic, that is, it does not contain any loops. On the other hand, P factorizes with respect to the undirected graph Gm which is given by the class of complete subsets D = {{v} ∪ Uv|v ∈ V }. This graph can be obtained from the directed graph G by completing all sets {v} ∪ Uv and then converting all directed edges into undirected ones.
    [Show full text]
  • Single-Source Shortest Paths (Ch
    Single-Source Shortest Paths (Ch. 24) Single-Source Shortest Paths The single-source shortest path problem (SSSP) Weights in SSSP algorithms include distances, times, hops, input: a directed graph G = (V, E) with edge weights, and a cost, etc. Used in lots of routing applications. specific source node s. goal: find a minimum weight (shortest) path from s to every other Note: BFS finds the shortest paths for the special case when all node in V (sometimes we only want the cost of these paths) edge weights are 1. Running time = O(V + E) It turns out that, in the worst case, finding the shortest path between s and a single The result of a SSSP algorithm can be viewed as a tree node t is no easier than finding the shortest paths between s and all other nodes rooted at s, containing a shortest (wrt weight) path from s à g = ¥ s to all other nodes. s à b = 2 2 3 s à c = 5 s à h = 7 a b c 2 3 5 6 a b 6 c s à d = 6 S 5 3 d 1 e S 3 s à e = 5 2 d 1 e 2 s à f = 3 f 4 g 2 h f 4 g 2 h 2 Negative-weight cycles Single-Source Shortest Paths Some graphs may have negative-weight cycles and these are Question: Can a shortest path contain a cycle? a problem for SSSP algorithms. What is the shortest path from a to d? No. b 9 · path a-c-e-d = -12+3-2 = -11 2 8 6 Suppose we have a shortest path p = áv0, v1, ..., vkñ and S -12 3 c = áv , v , ..., v ñ is a positive-weight cycle on p so that v = v · path a-c-e-d-a-c-e-d = a c e i i+1 j i j -12+3-2+(10-12+3-2)= -13 6 4 and w(c) > 0.
    [Show full text]
  • Algorithms (Winter 2021) Ungraded Problem Set 5 Due: Not for Submission Topic: Graphs: Basics + DFS
    CS 31: Algorithms (Winter 2021) Ungraded Problem Set 5 Due: Not for submission Topic: Graphs: Basics + DFS Problem 1 (Some operations with graphs). For the parts below, describe your algorithm summary in English and also provide a clear pseudocode. You may assume the graph G = (V; E) is input as adjacency lists. a. In this part the graph G is directed. You need to return the reverse graph Grev where all edges have been reversed. Grev should also be represented as adjacency lists. What is the runtime of your algo- rithm? K b. Recall the definition of a walk from vertex u to vertex v and a path from vertex u to v. Design an algorithm which takes input a walk from u to v, and returns a path from u to v. K Problem 2 (DFS firsts and lasts). For each of the following assertions, prove or give a counterexample. The graph G, unless mentioned, is directed. When you give a counter-example, you should explain why it is a counter-example. (a) If G contains a path from u to v and if first[u] < first[v] in a depth-first search of G, then last[v] < last[u]. K (b) If (u=v1; v2; : : : ; vk =v) is a path from u to v in G such that last[vi] < last[v] for all 1 ≤ i < k, then first[v] < first[vi] for all 1 ≤ i < k. K (c) If we run depth-first search on G in an arbitrary order to obtain first’s and last’s, and let σ be the increasing order of last[v]’s.
    [Show full text]