bfs and dfs bfs and dfs in directed graphs bfs in undirected graphs an improved undirected...

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  • Slide 1
  • BFS and DFS BFS and DFS in directed graphs BFS in undirected graphs An improved undirected BFS-algorithm
  • Slide 2
  • The Buffered Repository Tree (BRT) Stores key-value pairs (k,v) Supported operations: I NSERT (k,v) inserts a new pair (k,v) into T E XTRACT (k) extracts all pairs with key k Complexity: I NSERT : O((1/B)log 2 (N/B)) amortized E XTRACT : O(log 2 (N/B) + K/B) amortized (K = number of reported elements)
  • Slide 3
  • The Buffered Repository Tree (BRT) (2,4)-tree Leaves store between B/4 and B elements Internal nodes have buffers of size B Root in main memory, rest on disk Main memory Disk
  • Slide 4
  • Main memory Disk I NSERT (k,v) O(X/B) I/Os to empty buffer of size X B Amortized charge per element and level: O(1/B) Height of tree: O(log 2 (N/B)) Insertion cost: O((1/B)log 2 (N/B)) amortized Main memory Disk
  • Slide 5
  • Elements with key k E XTRACT (k) Number of traversed nodes: O(log 2 (N/B) + K/B) I/Os per node: O(1) Cost of operation: O(log 2 (N/B) + K/B) But careful with removal of extracted elements Main memory Disk Main memory Disk
  • Slide 6
  • Cost of Rebalancing O(N/B) leaf creations and deletions O(N/B) node splits, fusions, merges Each such operation costs O(1) I/Os O(N/B) I/Os for rebalancing Theorem: The BRT supports I NSERT and E XTRACT operations in O((1/B)log 2 (N/B)) and O(log 2 (N/B) + K/B) I/Os amortized.
  • Slide 7
  • Directed DFS Algorithm proceeds as internal memory algorithm: Use stack to determine order in which vertices are visited For current vertex v: Find unvisited out-neighbor w Push w on the stack Continue search at w If no unvisited out-neighbor exists Remove v from stack Continue search at vs parent Stack operations cost O(N/B) I/Os Problem: Finding an unvisited vertex
  • Slide 8
  • Directed DFS Data structures: BRT T Stores directed edges (v,w) with key v Priority queues P(v), one per vertex Stores unexplored out-edges of v Invariant: Not in P(v) In P(v) and in T In P(v), but not in T
  • Slide 9
  • Directed DFS Finding next vertex after vertex v: v E XTRACT (v): Retrieve red edges from T Remove these edges from P(v) using D ELETE Retrieve next edge using D ELETE M IN on P(v) Insert in-edges of w into T w Push w on the stack O(log 2 (|E|/B) + K 1 /B) O(sort(K 1 )) O(1 + (K 2 /B)log 2 (|E|/B)) O(1/B) amortized O((1/B)log m (|E|/B)) O(|V| log 2 (|E|/B) + |E|/B) O(|V| + sort(|E|)) O((|E|/B)log 2 (|E|/B)) O(|V|/B) O(sort(|E|)) Total: O((|V| + |E|/B)log 2 (|E|/B))
  • Slide 10
  • Directed DFS + BFS BFS can be solved using same algorithm Only modification: Use queue (FIFO) instead of stack Theorem: Depth first-search and breadth-first search in a directed graph G = (V,E) can be solved in O((|V|+|E|/B)log 2 (|E|/B)) I/Os. Exercise: Convince yourself that the priority queues P(v) are not necessary in the case of BFS.
  • Slide 11
  • Undirected BFS Observation: For v L(i), all its neighbors are in L(i 1) L(i) L(i + 1). Build BFS-tree level by level: Initially, L(0) = {r} Given levels L(i 1) and L(i): Let X(i) = set of all neighbors of vertices in L(i) Let L(i + 1) = X(i) \ (L(i 1) L(i)) Partition graph into levels L(0), L(1),... around source: L(0), L(1), L(2), L(3)
  • Slide 12
  • Undirected BFS Constructing L(i + 1): Retrieve adjacency lists of vertices in L(i) X(i) Sort X(i) Scan L(i 1), L(i), and X(i) to Remove duplicates from X(i) Compute X(i) \ (L(i 1) L(i)) Complexity: O(|L(i)| + sort(|L(i 1)| + |X(i)|)) I/Os O( ) I/Os|V| +sort(|E|) Theorem: Breadth-first search in an undirected graph G = (V,E) can be solved in O(|V| + sort(|E|)) I/Os.
  • Slide 13
  • A Faster BFS-Algorithm Problem with simple BFS-algorithm: Random accesses to retrieve adjacency lists Idea for a faster algorithm: Load more than one adjacency list at a time Reduces number of random accesses Causes edges to be involved in more than one iteration of the algorithm Trade-off
  • Slide 14
  • A Faster BFS-Algorithm (Randomized) Let 0 < < 1 be a parameter (specified later) Two phases: Build |V| disjoint clusters of diameter O(1/ ) Perform modified version of S IMPLE B FS Clusters C 1,...,C q formed using BFS from randomly chosen set V = {r 1,...,r q } of masters Vertex is chosen as a master with probability (coin flip) Observation: E[|V|] = |V|. That is, the expected number of clusters is |V|.
  • Slide 15
  • Forming Clusters (Randomized) Apply S IMPLE B FS to form clusters L(0) = V v C i if v is descendant of r i s
  • Slide 16
  • Forming Clusters (Randomized) Lemma: The expected diameter of a cluster is 2/ . E[k] 1/ Corollary: The clusters are formed in expected O((1/ )sort(|E|)) I/Os. x v1v1 v2v2 v3v3 v4v4 v5v5 s vkvk
  • Slide 17
  • Forming Clusters (Randomized) Form files F 1,...,F q, one per cluster F i = concatenation of adjacency lists of vertices in C i Augment every edge (v,w) F i with the start position of file F j s.t. w C j : Edge = triple (v,w,p j ) s
  • Slide 18
  • The BFS-Phase Maintain a sorted pool H of edges s.t. adjacency lists of vertices in L(i) are contained in H Scan L(i) and H to find vertices in L(i) whose adjacency lists are not in H Form list of start positions of files containing these adjacency lists and remove duplicates Retrieve files, sort them, and merge resulting list H with H Scan L(i) and H to build X(i) Construct L(i + 1) from L(i 1), L(i), and X(i) as before O((|L(i)| + |H|)/B) O(sort(|L(i)|)) O(K + sort(|H|) + |H|/B) O((|L(i)| + |H|)/B) O(sort(|L(i)| + |L(i 1)| + |X(i)|))
  • Slide 19
  • The BFS-Phase I/O-complexity of single step: O(K + |H|/B + sort(|H| + |L(i 1)| + |L(i)| + |X(i)|)) Expected I/O-complexity: O( |V| + |E|/( B) + sort(|E|)) Choose Theorem: BFS in an undirected graph G = (V,E) can be solved in I/Os.
  • Slide 20
  • Single Source Shortest Paths The tournament tree SSSP in undirected graphs SSSP in planar graphs
  • Slide 21
  • Single Source Shortest Paths Need: I/O-efficient priority queue I/O-efficient method to update only unvisited vertices
  • Slide 22
  • The Tournament Tree =I/O-efficient priority queue Supports: I NSERT (x,p) D ELETE (x) D ELETE M IN D ECREASE K EY (x,p) All operations take O((1/B)log 2 (N/B)) I/Os amortized Note: N = size of the universe # elements in the tree
  • Slide 23
  • The Tournament Tree Static binary tree over all elements in the universe Elements map to leaves, M elements per leaf Internal nodes have signal buffers of size M Root in main memory, rest on disk Main memory Disk Internal nodes store between M/2 and M elements
  • Slide 24
  • Main memory Disk The Tournament Tree Elements stored at each node are sorted by priority Elements at node v have smaller priority than elements at vs descendants Convention: x T if and only if p(x) is finite
  • Slide 25
  • The Tournament Tree Deletions Operation D ELETE (x) signal D ELETE (x) x D ELETE (x) U PDATE (x, ) v
  • Slide 26
  • The Tournament Tree Insertions and Updates Operations I NSERT (x,p) and D ECREASE K EY (x,p) signal U PDATE (x,p) x w v Current priority p If p < p: Update If p p: Do nothing All elements < p Forward signal to w At least one element p Insert x Send D ELETE (x) to w
  • Slide 27
  • The Tournament Tree Handling Overflow Let y be element with highest priority p y Send signal P USH (y,p y ) to appropriate child of v y w v
  • Slide 28
  • The Tournament Tree Keeping the Nodes Filled w v O(M/B) I/Os to move M/2 elements one level up the tree
  • Slide 29
  • Main memory Disk The Tournament Tree Signal Propagation Scan vs signal, partition into sets X u and X w Load u into memory, apply signals in X u to u, insert signals into us signal buffer Do the same for w O((|X| + M)/B) = O(|X|/B) I/Os
  • Slide 30
  • The Tournament Tree Analysis Elements travel up the tree Cost: O(1/B) I/Os amortized per element and level O((K/B)log 2 (N/B)) I/Os for K operations Signals travel down the tree Cost: O(1/B) I/Os amortized per signal and level O(K) signals for K operations O((K/B)log 2 (N/B)) I/Os Theorem: The tournament tree supports I NSERT, D ELETE, D ELETE M IN, and D ECREASE K EY operations in O((1/B)log 2 (N/B)) I/Os amortized.
  • Slide 31
  • Single Source Shortest Paths Modified Dijkstra: Retrieve next vertex v from priority queue Q using D ELETE M IN Retrieve vs adjacency list Update distances of all of vs neighbors, except predecessor u on the path from s to v Repeat O(|V| + (E/B)log 2 (V/B)) I/Os using tournament tree
  • Slide 32
  • Single Source Shortest Paths Problem: Observation: If v performs a spurious update of u, u has tried to update v before. Record this update attempt of u on v by insterting u into another priority queue Q Priority: d(s,u) + w({u,v}) u v
  • Slide 33
  • Single Source Shortest Paths Second modification: Retrieve next vertex using two D ELETE M IN s, one on Q, one on Q Let (x,p x ) be the element retrieved from Q, let (y,p y ) be the element retrieved from Q If p x p y : re-insert (y,p y ) into Q and proceed as normal If p x < p y : re-insert (x,p x ) into Q and perform a D ELETE (y) on Q
  • Slide 34
  • Single Source Shortest

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