Lecture 19 Traversal Algorithm

1
Lecture 19 Traversal Algorithm

• July 28th, 2003

2
Euler Path Hamilton Circuit

• Sectn 6.2 exhibits these traversals (ones of
  classic(al) status)
• Euler Encounter all nodes BUT use each arc
  (only) one time --- do not have to return to
  starting point (see immedly below)
• An Euler Path (p. 427) exists in a connected
  graph iff there are
• -- no odd nodes --- (start end at
  same Node) or
• -- two odd nodes --- (start at one
  Node end at another)
• Hamilton A cycle (circuit) that touches all
  the nodes but is otherwise free
• A Hamiltonian Circuit problem has no general
  solution
• --- ie. No efficient algorithm exists
• --- A complete graph (defd on p. 346)
  as one in which any two distinct nodes
  are adjacent) for n gt 2 represents a special
  case and
• there are others besides!

3
Shortest Path Minimal Spanning Tree

• Have to short-circuit in Su 04
• Along with Euler Path Ham Circuit
• GOT COMPANY may not have been covered in recent
  Summers?

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What is Traversal --- Sectn 6.4 pp. 447ff

• Graph traversal , a process to retrace arcs so
  that each node of the graph is visited.
• Recall from Section 5.2, We have several tree
  traversal algorithms
• Two general traversal (search) methods
• Depth first search
• Breadth first search
• Traversals (also) solve path questions about a
  graph (some already seen)
• Is there a path from node x to node y?
• Is there a path through G that use each arc once?
  
• What is the minimum weight path between x and y ?

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Traversal --Example

• Fig 6.13 p. 447B We can generalize tree
  search (We deal with Depth First Search re an
  algo and text descriptions --- in the next two
  slides)

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Depth-First Search (Algo)

• DepthFirst (graph G node a)
• mark as visited
• write(a)
• for each node n adjacent to a do
• if n not visited then
• DepthFirst(G,n)
• end if
• end for
• end DepthFirst

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Depth-First Search (Algo)

• (Review What idea have we seen b4 in cases like
  this? Recursion)
• In depth-first search
• --- We begin at an arbitrary node a of the
  graph --- We mark it visited, and write
  it down.
• --- We then strike out on a path away from a,
  visitg and writing down nodes,
  proceeding as far as possible until there are no
  more unvisited nodes on that path.
• --- We then back up the path we took down and
  at each node explore any new side paths, until
  finally we retreat back to a.

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Depth First Search (Examples)

• Ex. 11 p. 448 What is the depth-first
  traversal (starting from a)? (You kinda have to
  trace it out with your fingers on the book
  page)
• Ans. a, b, c, d, f, g, e, h, I, k, j NODE I
  ????
• Prax 14 p. 449 What is the (depth-first)
  traversal starting from a? (Again, traceg by
  hand is called for.)
• Bk Correction is needed - the graph is missg its
  right most node (it is really l).
• Ans. a, e, d, b, c, i, f, g, h, l, k, m, j

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Algorithm Breadth-first search

• BreadthFirst (graph G node a)
• Local variable queue of nodes Q
• initialize Q to be empty
• mark a visited
• write(a)
• enqueue(a,Q)
• while Q is not empty do
• for each node n adjacent to front(Q) do
• if n not visited then
• mark n visited
• write(n)
• enqueue(n,Q)
• end if
• end for
• dequeue(Q)
• end while
• end BreadthFirst

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Breadth-first Search

• In breadth-first search
• --- we begin at an arbitrary node a
• --- we first fan out from node a to visit
  nodes that are adjacent to a --- in a
• left to right fashion
• --- then we fan out from those nodes.
• A queue structure is required here.

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Ex. 12 p. 451 Breadth-first Search Example

• What is in the (required) queue?
• (See the details on p. 451-52 folog the
  algo pseudocode)
• a
• b,e,h,I See Fig. 6.15, p. 450
• e,h,i,c after we remove b and
  add c
• See Fig.
  6.16, p. 452
• h,i,c,j after we remove e and
  add j
• j,k,d,f
• f,g and we are now finished
  
• Ans a b e h i c j k d f g
• Prax 15 p. 452 asks us to do BREADTH 1st to
  the same case we did in Prax 14 as a Depth First
  example

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Time complexity?

• Given a graph with n nodes and m arcs.
• Which (kind of) graph rep. is better?
• adjacency list
• or
• adjacency matrix?
• (The ans. is the former! --- at least in
  many cases (most applic cases?)
• What is the time complexity (both DF BF)?
• O(max(m,n))
• (Note to note maker --- Should we use ? instead
  of O here (compare book closer) and continue to
  search for what seems to scant opps. to get
  symbology up to snuff?)

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More applications

• These provide thots about/for performg other
  graph-related tasks, such as
• An efficient algorithm for reachability in
  directed graphs
• Consider as cases
• Sparse graphs, O(n2)
• Non sparse graphs, O(n3) --- compare gen matrix
  mult

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More applications/cont.d

• Topological sort ---
• In Sectn 4.2 topolog sort is related to a
  problem of determining a total order from a Poset
  --- a comment that seems only to serve to
  telegraph our startg point.
• Prax 17 (p 454)
  a c
• 1. Start at a do DEPTH-1st Search
  \ / \
• to e, say, via b c (Put (1) at e) (2)
  at c b e
• (Puts occur when we make a final
  \ /
• stop at a node before we back-up)
  d
• 2. Back up to b proceed to d stop a last
  time put (3) on
• d and ditto with resp. b and, yes, a,
  too
• 3. Unravel from high (5) to low (1)
  Ans. a b d c e
• 4. (See possible ans. a b c d e !!!)

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More applications/cont.d

• Prolog, when processing a query based on a
  recursive definition, purses a depth-first
  search.
• (See Ex 40, Ch 1)
• Consider our natural language (syntax example)
  and realize the processing order

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Exercise

• Exercise 6.4
• 1,2,3,11,12,13