EMIS 8374 Search Algorithms Updated 9 February 2004

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EMIS 8374Search AlgorithmsUpdated 9 February
2004
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Generic Search Algorithm (pg 74)

• Find all nodes reachable via a directed path from
  source node s
• All nodes are either marked or unmarked
• Arc (i,j) is admissible if i is marked and j is
  not marked and inadmissible, otherwise.

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Initialization

• unmark all nodes in N
• mark source node s
• pred(s) 0
• next 1
• order(s) 1
• LIST s

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Main Loop

• while LIST is not empty do
• begin
• select a node i from LIST
• if there is an admissible arc (i, j) then
• begin
• mark node j pred(j) i, next next 1,
• order(j) next add node j to LIST
• end
• else delete node i from LIST
• end

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Search Example 1 s 4
2
4
1
6
3
5
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Search Example 1 Initialization
2
4
1
6
3
5
next 1 LIST 4 pred4 0 order4 1
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Search Example 1 While Loop
2
4
1
6
3
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next 1 LIST 4 pred4 0 order4 1
i 4. Find an admissible arc.
j 6.
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Search Example 1 Mark Node 6
2
4
1
6
3
5
next 2 LIST 4, 6 pred6 4 order6 2
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Search Example 1 While Loop
2
4
1
6
3
5
LIST 4, 6. i 6. Find an admissible arc. LIST
4.
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Search Example 1 While Loop
2
4
1
6
3
5
LIST 4
i 4. Find an admissible arc.
j 3.
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Search Example 1 Mark Node 3
2
4
1
6
3
5
next 3 LIST 4, 3 pred3 4 order3 3
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Search Example 1 While Loop
2
4
1
6
3
5
LIST 4, 3. i 4. Find an admissible arc. LIST
3.
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Search Example 1 While Loop
2
4
1
6
3
5
LIST 3. i 3. Find an admissible arc.
j 5.
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Search Example 1 Mark Node 5
2
4
1
6
3
5
next 4 LIST 5, 3 pred5 3 order5 4
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Search Example 1 While Loop
2
4
1
6
3
5
LIST 5, 3. i 3. Find an admissible arc. LIST
5.
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Search Example 1 While Loop
2
4
1
6
3
5
LIST 5. i 5. Find an admissible arc. LIST
.
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Search Example 1 Search Tree
1
2
4
2
1
6
4
3
3
5
order3 3, order4 1, order5 4, order
6 2
Pred4 0,
Pred6 1,
Pred5 3.
Pred3 4
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Trees

• An undirected graph is connected if there is a
  path between an pair of nodes
• A tree T(N,A) is a connected graph with with n-1
  arcs (edges)
• A graph with n nodes must have at least n-1 arcs
  to be connected

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A Connected Graph
2
4
1
6
3
5
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A Tree
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Complexity of Generic Search

• Each iteration of the While loop either adds a
  new node to LIST or removes a node from LIST.
• A given node can be added at most once and
  removed at most once.
• Thus, the loop runs at most 2n times.

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Complexity of Generic Search Work per iteration
of the While Loop

• Selecting a node in LIST is O(1).
• Deleting a node from LIST is O(1).
• Marking a node (updating pred, next, and order,
  etc) is O(1).
• Therefore, the work per iteration is O(1) work
  required finding an admissible arc.

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Finding an Admissible Arc Incident to Node i
Naïve Approach

• found 0
• for (i, j) in A(i)
• if j is unmarked then
• begin
• found 1
• break
• end
• if found 1 then
• mark node j pred(j) i etc
• else delete node i from LIST

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Finding an Admissible Arc Incident to Node i
Naïve Approach

• Worst-Case
• when there are no admissible arcs incident to
  node i this approach will check all of node is
  neighbors to see if they are marked.
• O(n)
• Overall complexity of generic search 2n x n
  O(n2)

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Finding an Admissible Arc Incident to Node i
Efficient Approach

• Use an adjacency list representation of the
  network
• Maintain a current arc data structure for each
  node.
• Indicates the next candidate arc in the adjacency
  list for node i.
• The next time node i is selected from LIST check
  the current arc first.
• If the current arc points to a marked node we
  move to the next arc in the adjacency list.
• If the we reach the end of the adjacency list for
  a given node we remove it from LIST.

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Adjacency List Implementation
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Search from Node 4 LIST 4
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Mark Node 6
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Advance Current Arc for Node 4
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LIST 4, 6, i 6
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LIST 4, i 4, mark 3
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LIST 4,3, i 4
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LIST 3, i 3, mark 5
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LIST 3, 5, i 3
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LIST 5, i 5
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LIST
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Complexity with Adjacency List

• For each node i marked by the search, we check
  update/test the current arc parameter exactly
  once for each arc in A(i).
• The total number of operations required by the
  search algorithm to find admissible arcs is
  proportional to the sum of A(i) for all the
  nodes it marks.
• Overall complexity is O(n) for marking nodes,
  updating pred vector etc, plus O(m) for finding
  admissible arcs O(n m) O(m)

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Breadth-First Search

• Implement LIST as an ordered list
• Always pick the first node in the list
• Always add newly marked nodes to the end of the
  list
• Level parameter
• Initialize level(s) 0
• Set level(i) level(predi) 1
• Level(i) number of arcs in path from s to i

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Breadth-First Search s 1
2
4
1
6
1
3
5
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Breadth-First Search s 1
2
2
4
1
6
1
3
5
LIST 1, 2
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Breadth-First Search s 1
2
2
4
1
6
1
3
5
3
LIST 1, 2, 3
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Breadth-First Search s 1
2
2
4
1
6
1
3
5
3
LIST 2, 3
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Breadth-First Search s 1
2
2
4
1
6
1
3
5
3
LIST 2, 3
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Breadth-First Search s 1
2
4
2
4
1
6
1
3
5
3
LIST 2, 3, 4
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Breadth-First Search s 1
2
4
2
4
1
6
1
3
5
3
5
LIST 2, 3, 4, 5
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Breadth-First Search s 1
2
4
2
4
1
6
1
3
5
3
5
LIST 3, 4, 5
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Breadth-First Search s 1
2
4
2
4
1
6
1
3
5
3
5
LIST 4, 5
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Breadth-First Search s 1
2
4
2
4
6
1
6
1
3
5
3
5
LIST 4, 5, 6
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Breadth-First Search s 1
2
4
2
4
6
1
6
1
3
5
3
5
LIST 5, 6
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Breadth-First Search s 1
2
4
2
4
6
1
6
1
3
5
3
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LIST 6
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Breadth-First Search s 1
2
4
2
4
6
1
6
1
3
5
3
5
LIST
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Problems Solved by BFS

• Shortest Path problem when all arcs have the same
  cost.
• s-t shortest path
• all-pairs shortest path problem
• Determining strong connectivity
• Is there a directed path between all pairs of
  nodes?

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Levels of a BFS Tree

• Root node is at level 0
• Descendants of the root are at level 1
• IF node i is at level k Then
• the children of i are at level k1

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BFS Solves s-t Shortest Path
2
4
1
6
3
5
Level 0
Level 1
Level 2
Level 3
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Unweighted Shortest Path Problems

• cij 1 for all (i,j)
• BFS solves s-t shortest path problem in O(m)
  time.
• BFS solves all-pairs shortest path problem in
  O(nm) time.
• BFS determines strong connectivity in O(nm) time.

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Depth-First Search

• Always pick the last (most recently added) node
  in the LIST
• Pick marked nodes in a last-in, first-out order
  (LIFO)
• Breadth-First Search uses a FIFO order
• Makes a deep probe, creating as long a path as
  possible
• Backs up when it cant mark a new node

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Depth-First Search
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Depth-First Search
5
2
4
List
List 1
List 1
List 1,2,5,6
List 1,2
6
List 1,2
List 1,2
3
List 1,2,5
List 1,2,4
List 1,2,5
List 1,2,4
List 1,2,4,3
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Depth-First Search Tree
5
2
2
4
1
6
1
4
3
5
6
3
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Properties of a DFS Tree

• If node j is a descendant of node i, then
  order(j) gt order(i)
• All descendants of any node are ordered
  consecutively