CSE 502N Fundamentals of Computer Science

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CSE 502NFundamentals of Computer Science

• Fall 2004
• Lecture 16
• Breadth-first Search (CLRS 22.2)
• Dijkstras Shortest Path (CLRS 24.3)

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Breadth-first search

• Basis for Prims minimum-spanning tree and
  Dijkstras single-source shortest paths algorithm
• Given a graph G (V,E) and a source vertex s,
  breadth-first search computes the smallest number
  of edges from s to each reachable vertex
• G may be directed or un-directed
• Produces a breadth-first tree with root s that
  contains all reachable vertices
• Path from s to reachable vertex v in tree
  corresponds to shortest path from s to v in G
• BFS constructs the breadth-first tree by
  discovering all reachable vertices at the current
  distance k before moving to vertices at distance
  k 1

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Breadth-first search

• Assign colors to vertices to denote status
• White undiscovered vertex
• Gray discovered vertex with unexplored adjacency
  list
• Black discovered vertex, explored adjacency list
• All vertices adjacent to a black vertex are
  discovered
• Gray vertices may have adjacent white vertices
• Frontier between discovered and undiscovered
  vertices
• When a white vertex v is discovered while
  scanning the adjacency list of previously
  discovered vertex u, vertex v and edge (u,v) are
  added to the breadth-first tree
• Vertex u is the predecessor or parent of v in the
  treepv u

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BFS Pseudocode
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BFS Analysis

• Loop invariant the queue consists of the set of
  gray vertices
• Each node is enqueued and dequeued at most once
• Queuing operations are O(1), thus O(V) time is
  spent on queuing operations
• Each adjacency list is scanned at most once
• Sum of lengths of adjacency lists is Q(E), time
  spent scanning is O(E)
• Initialization overhead is O(V)
• Total running time is O(VE)

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Single-source shortest paths

• Given a weighted, directed graph G (V,E), find
  the shortest path from a given source vertex s Î
  V to each vertex v Î V
• Also can be used to solve the single-destination
  and single-pair shortest-path problems
• Let w(u,v) be the real valued weight of edge
  (u,v)
• Let w(p) be the weight of path p áv1,v2,,vkñ
• Equal to the sum of the weights of its
  constituent edges
• Let d(u,v) be the shortest-path weight from u to
  v
• A shortest path from u to v is any path p with
  weightw(p) d(u,v)

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Optimal substructure

• Most shortest-path algorithms rely on the
  property that a shortest path between two
  vertices contains other shortest paths within it
• Given a weighted, directed graph G (V,E) with
  real valued edge weights
• Let p áv1,v2,,vkñ be a shortest path from
  vertex v1 to vertex vk
• For any i and j such that 1 i j k, let pij
  ávi,vi1,,vjñ be the subpath of p from vertex
  vi to vertex vj
• Then, pij is a shortest path from vi to vj

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Relaxation

• For each vertex v Î V, we maintain a
  shortest-path estimate dv which is an upper
  bound on the weight of a shortest path from s to
  v
• Initialize shortest-path estimates and
  predecessors in time Q(V)
• Relaxing an edge (u,v) consists of testing
  whether the current shortest path to v can be
  improved by going through u
• If so, dv and pv are updated
• Shortest-path algorithms call Initialize-Single-So
  urce then repeatedly relax edges
• Relaxation is the only means by which
  shortest-path estimates and predecessors change

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Dijkstras algorithm

• Solves the single-source shortest-paths problem
  on weighted, directed graph with nonnegative edge
  weights
• Maintain a set S of vertices whose final
  shortest-path weights from the source s have been
  determined
• Algorithm repeats the following steps until all
  reachable vertices have been added to S
• Select the vertexu Î V S with
  minimumshortest-path estimate
• Add u to S
• Relax all edges leaving u
• Uses a min-priority queue Q keyed by
  shortest-path estimates