Multiple-Source Shortest Paths in Planar Graphs Allowing Negative Lengths

1
Multiple-Source Shortest Paths in Planar Graphs
Allowing Negative Lengths

• Philip Klein
• Brown University

2
Main Result

• For plane graph with boundary r1,,rs,O(n log n)
  algorithm to find all shortest-path trees rooted
  at the ris.

3
Main Result

• For plane graph with boundary r1,,rs, O(n log
  n) algorithm to find all shortest-path trees
  rooted at the ris. implicit representation

4
Main Result

• For plane graph with boundary r1,,rs, O(n log
  n) algorithm to find all shortest-path trees
  rooted at the ris. implicit representation

• Negative lengths are ok

5
Main Result

• For plane graph with boundary r1,,rs,O(n log n)
  algorithm to find all shortest-path trees rooted
  at the ris. implicit representation

• Negative lengths are ok
• Given k source-sink pairs (where each source is
  an ri), can find corresponding distances in O((n
  k) log n) time

6
Some Applications

• Single-source shortest-path tree in planar graph
  with some negative lengths O(n log n).
  Previous bound O(n log3 n) Fakcharoenphol,
  Rao
• Testing feasibility of planar flow with multiple
  sources and sinks, and upper and lower
  capacities O(n log n).
• Finding a minimum ratio cut in a planar graph
  (used in, e.g., segmentation of images Cox, Rao,
  Zhong)
• O(n log n log CW) where C sum of costs, W sum
  of weights.

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Some Applications

• Preprocessing a planar graph to admit O(
  )-time exact distance queries O(n log2 n).
  Previous bound Fakch., Rao O(n log3 n)
• Preprocessing undirected planar graph to admit
  O(e-1)-time approx. distance queries O(e-1n log2
  n). Previous bound Thorup O(e-2n log3 n)
• (similar improvement for directed graphs)
• Slightly faster dynamic algorithms for
  planar-graph distances (improving on Fakch.,
  Rao)

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Previous Work
Single-source shortest paths in planar graphs
with negative lengths
Lipton, Rose, Tarjan, 1979 O(n1.5)
Henzinger, Klein, Rao, Subramanian, 1997 O(n4/3 log (nL))
Fakcharoenphol, Rao, 2001 O(n log3 n)
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Previous Work
Multiple-source shortest paths in planar graphs
Schmidt, 1995 For special case of n-node grid
DAG, after O(n log n) preprocessing, can find
distance from any node in column 0 to any other
node in O(log n) time.
10
Talk Overview

• Analysis technique O(n) representation of
  multiple-source shortest-path trees
• Shortest-path-update technique represent partial
  solution by spanning tree, not by distance
  labeling.
• Unrelaxed-edge-selection technique use dynamic
  tree in planar dual
• Partial-solution-tree property Each path P in
  tree is the shortest path to the right of P.

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Analysis Technique

• Nodes along boundary of graph

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Analysis Technique

• Fix a node at v

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Analysis Technique

• Shortest paths to v dont cross

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Analysis Technique

• For each edge entering v, paths using the edge
  start from consecutive ris.

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Analysis Technique

• As we iterate through nodes r1,,rs,edge used to
  reach v rarely changes
• Number of changes indegree(v)

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Analysis Technique

• As we iterate through nodes r1,,rs,edge used to
  reach v rarely changes
• Number of changes indegree(v)

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Analysis Technique

• As we iterate through nodes r1,,rs,edge used to
  reach v rarely changes
• Number of changes indegree(v)

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Analysis Technique

• As we iterate through nodes r1,,rs,edge used to
  reach v rarely changes
• Number of changes indegree(v)

19
Analysis Technique

• As we iterate through nodes r1,,rs,edge used to
  reach v rarely changes
• Number of changes indegree(v)

20
Analysis Technique

• As we iterate through nodes r1,,rs,edge used to
  reach v rarely changes
• Number of changes indegree(v)

21
Analysis Technique

• As we iterate through nodes r1,,rs,edge used to
  reach v rarely changes
• Number of changes indegree(v)

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Analysis Technique

• As we iterate through nodes r1,,rs,edge used to
  reach v rarely changes
• Number of changes indegree(v)

23
Analysis Technique

• As we iterate through nodes r1,,rs,edge used to
  reach v rarely changes
• Number of changes indegree(v)

24
Analysis Technique

• As we iterate through nodes r1,,rs,edge used to
  reach v rarely changes
• Number of changes indegree(v)

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Analysis Technique

• Now consider shortest-path trees from r1,,rs.

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Analysis Technique

• Number of changes as we iterate through ris is
• indegree(v)

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Analysis Technique

• Number of changes as we iterate through ris is
• indegree(v), which is number of edges

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Traditional Shortest-path Algorithm

• Maintains an assignment d(.) of distance
  estimates to nodes
• An edge uv is unrelaxed if
• d(u) len(uv) lt d(v)

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Traditional Shortest-path Algorithm

• Maintains an assignment d(.) of distance
  estimates to nodes
• An edge uv is unrelaxed if
• d(u) len(uv) lt d(v)
• Length of a path from root to v through relaxed
  edges is at least d(v).
• To get shorter path to v, need to use at least
  one unrelaxed edge.

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Traditional Shortest-path Algorithm

• Maintains an assignment d(.) of distance
  estimates to nodes
• An edge uv is unrelaxed if
• d(u) len(uv) lt d(v)
• Relaxing edge uv means assigning d(u) len(uv)
  to d(v)

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Traditional Shortest-path Algorithm

• Maintains an assignment d(.) of distance
  estimates to nodes
• An edge uv is unrelaxed if
• d(u) len(uv) lt d(v)
• Relaxing edge uv means assigning d(u) len(uv)
  to d(v)

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Traditional Shortest-path Algorithm

• Maintains an assignment d(.) of distance
  estimates to nodes
• An edge uv is unrelaxed if
• d(u) len(uv) lt d(v)
• Relaxing edge uv means assigning d(u) len(uv)
  to d(v)
• After this assignment, uv is relaxed but edges vw
  out of v might now be unrelaxed.
• Updates to d(.) propagate slowly

33
Traditional Shortest-path Algorithm

• Maintains an assignment d(.) of distance
  estimates to nodes
• An edge uv is unrelaxed if
• d(u) len(uv) lt d(v)
• Relaxing edge uv means assigning d(u) len(uv)
  to d(v)
• After this assignment, uv is relaxed but edges vw
  out of v might now be unrelaxed.
• Updates to d(.) propagate slowly

34
New Approach

• Maintain a tentative shortest-path tree T rooted
  at r
• Each node except r has a parent edge in T
• Tree defines an assignment dT(.)
• dT(v) length of r-to-v path in T
• An edge uv is unrelaxed if
• dT(u) len(uv) lt dT(v)
• Relaxing an edge uv means replacing vs parent
  edge in T with uv
• Values of dT(.) automatically updated

35
New Approach

• Maintain a tentative shortest-path tree T rooted
  at r
• Each node except r has a parent edge in T
• Tree defines an assignment dT(.)
• dT(v) length of r-to-v path in T
• An edge uv is unrelaxed if
• dT(u) len(uv) lt dT(v)
• Relaxing an edge uv means replacing vs parent
  edge in T with uv
• Values of dT(.) automatically updated

36
Generic Single-Source Algorithm

• Start with some tree rooted at r.
• Repeat
• select an unrelaxed edge
• relax it
• Until none exist
• Questions
• Which tree to start with?
• How to find an unrelaxed edge?
• How much time is required?

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Generic Single-Source Algorithm

• Start with some tree rooted at r.
• Repeat
• select an unrelaxed edge
• relax it
• Until none exist
• Questions
• Which tree to start with?
• How to select an unrelaxed edge?
• How much time is required?

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Questions for Single-source Algorithm A Preview

• How to select an unrelaxed edge?
• Leafmost unrelaxed edge in dual spanning tree
• Use a dynamic tree to represent dual spanning
  treeO(log n) amortized time per relaxation step
• Which tree to start with?
• Rightmost-search tree
• How much time required?
• Specify invariant that guarantees O(n) relaxation
  steps

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Planar Dual

• Every planar embedded graph has a dual graph
• Dual nodes primal faces
• Dual edge cross primal edges

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Planar Dual

• Every planar embedded graph has a dual graph
• Dual nodes primal faces
• Dual edge cross primal edges
• For any primal spanning tree T

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Planar Dual

• Every planar embedded graph has a dual graph
• Dual nodes primal faces
• Dual edge cross primal edges
• For any primal spanning tree T, duals of edges
  not in T form a dual spanning tree T

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Planar Dual

• Every planar embedded graph has a dual graph
• Dual nodes primal faces
• Dual edge cross primal edges
• For any primal spanning tree T, duals of edges
  not in T form a dual spanning tree T

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Planar Dual

• Every planar embedded graph has a dual graph
• Dual nodes primal faces
• Dual edge cross primal edges
• For any primal spanning tree T, duals of edges
  not in T form a dual spanning tree T

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How to Select an Unrelaxed Edge

• Dual spanning tree T, rooted at infinite face

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How to Select an Unrelaxed Edge

• Dual spanning tree T, rooted at infinite face

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How to Select an Unrelaxed Edge

• Dual spanning tree T, rooted at infinite face
• Choose a leafmost unrelaxed edge
• (unrelaxed edge with no descendant unrelaxed edge)

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Choosing Unrelaxed Edge Implementation

• Use a dynamic tree data structure Sleator,
  Tarjan to represent the dual tree T.
• O(log n) time per operation
• Structure Operations
• Cut an edge, breaking a tree into two
• Join two trees, making the root of one into a
  child of some node of the other
• Evert a tree, changing its root

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Choosing Unrelaxed Edge Implementation

• Use a dynamic tree data structure Sleator,
  Tarjan to represent the dual tree T.
• O(log n) time per operation
• Structure Operations
• Cut an edge, breaking a tree into two
• Join two trees, making the root of one into a
  child of some node of the other
• Evert a tree, changing its root

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Choosing Unrelaxed Edge Implementation

• Use a dynamic tree data structure Sleator,
  Tarjan to represent the dual tree T.
• O(log n) time per operation
• Structure Operations
• Cut an edge, breaking a tree into two
• Join two trees, making the root of one into a
  child of some node of the other
• Evert a tree, changing its root

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Choosing Unrelaxed Edge Implementation

• Dynamic tree implicitly represents assignments of
  costs to nodes
• Cost operations
• search a v-to-root path for the minimum-cost node
• add a number ? to costs of all nodes in a
  v-to-root path
• We need some new features

51
Choosing Unrelaxed Edge Using Dynamic Tree

• For each non-tree primal edge uv, define
• s(uv) dT(u) len(uv) dT(v)
• Measure of how relaxed the edge is.
• Unrelaxed edges have negative s values

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Choosing Unrelaxed Edge Using Dynamic Tree

• For each non-tree primal edge uv, define
• s(uv) dT(u) len(uv) dT(v)
• Measure of how relaxed the edge is.
• Unrelaxed edges have negative s values
• Dual edges same s values as primal edges

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Choosing Unrelaxed Edge Using Dynamic Tree

• For each non-tree primal edge uv, define
• s(uv) dT(u) len(uv) dT(v)
• Measure of how relaxed the edge is.
• Unrelaxed edges have negative s values
• Dual edges same s values as primal edges
• Use dynamic tree to represent dual tree F
• Dynamic tree implicitly represents s values
• Modify dynamic tree to support O(log n) search
  for a leafmost edge with negative s value.
• Can update s values in O(log n) time after
  relaxation step.

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Modification of s Values in Dual Tree

• Relaxing edge wv reduces the value of dT(.) by
  -s(wv) for v and vs descendents in T.
• Replace vs parent edge uv in T with wv.

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Modification of s Values in Dual Tree

• Relaxing edge wv reduces the value of dT(.) by
  -s(wv) for v and vs descendents in T.
• Replace vs parent edge uv in T with wv.

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Modification of s Values in Dual Tree

• Relaxing edge wv reduces the value of dT(.) by
  -s(wv) for v and vs descendents in T.
• Replace vs parent edge uv in T with wv.

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Modification of s Values in Dual Tree

• Relaxing edge wv reduces the value of dT(.) by
  -s(wv) for v and vs descendents in T.
• Replace vs parent edge uv in T with wv.

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Modification of s Values in Dual Tree

• Relaxing edge wv reduces the value of dT(.) by
  -s(wv) for v and vs descendents in T.
• Replace vs parent edge uv in T with wv.

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Modification of s Values in Dual Tree

• Relaxing edge wv reduces the value of dT(.) by
  -s(wv) for v and vs descendents in T.
• Replace vs parent edge uv in T with wv.

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Modification of s Values in Dual Tree

• Relaxing edge wv reduces the value of dT(.) by
  -s(wv) for v and vs descendents in T.
• Replace vs parent edge uv in T with wv.
• Let xy dual edge corresp. to uv

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Modification of s Values in Dual Tree

• Relaxing edge wv reduces the value of dT(.) by
  -s(wv) for v and vs descendents in T.
• Replace vs parent edge uv in T with wv.
• Let xy dual edge corresp. to uv

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Modification of s Values in Dual Tree

• Relaxing edge wv reduces the value of dT(.) by
  -s(wv) for v and vs descendents in T.
• Replace vs parent edge uv in T with wv.
• Let xy dual edge corresp. to uv
• Must change s values along x-to-y path

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Modification of s Values in Dual Tree

• Orientation of dual edges reflect orientation of
  primal edges

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Modification of s Values in Dual Tree

• Orientation of dual edges reflect orientation of
  primal edges
• Keep track of orientation of dual edges in dual
  spanning tree

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Modification of s Values in Dual Tree

• Define dynamic-tree operation Given a node v
  and a number ?, changeValue(v,?) changes the s
  values of all edges e on the v-to-root path

? if e points towards root - ? if e
points away from root
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Modification of s Values in Dual Tree

• Define dynamic-tree operation Given a node v
  and a number ?, changeValue(v,?) changes the s
  values of all edges e on the v-to-root path
• To modify s values in dual tree
• Set ? s(wv)
• Call
• changeValue(x, ?)
• changeValue(y, -?)

? if e points towards root - ? if e
points away from root
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Structural Changes in Dual Tree

• After modifying s values, use cut, evert, and
  join to change dual tree when relaxing an edge.

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Structural Changes in Dual Tree

• After modifying s values, use cut, evert, and
  join to change dual tree when relaxing an edge.

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Structural Changes in Dual Tree

• After modifying s values, use cut, evert, and
  join to change dual tree when relaxing an edge.

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Structural Changes in Dual Tree

• After modifying s values, use cut, evert, and
  join to change dual tree when relaxing an edge.

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Single-Source Algorithm

• Start with some tree rooted at r.
• Repeat
• select a leafmost unrelaxed edge
• relax it
• until none exist
• Questions
• Which tree to start with? (Later.)
• How to find a leafmost edge? Use the dual
  dynamic tree.
• How to bound time required? Each relaxation
  requires amortized O(log n) time. Need to bound
  number of relaxations.

72
Single-Source Algorithm

• Start with some tree rooted at r.
• Repeat
• select a leafmost unrelaxed edge
• relax it
• until none exist
• Questions
• Which tree to start with? (Later.)
• How to find a leafmost edge? Use the dual
  dynamic tree.
• How to bound time required? Each relaxation
  requires amortized O(log n) time. Need to bound
  number of relaxations.
• Theorem Each edge is relaxed at most once.
  (Proof later)

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Notation Tv

• For a tree T and a node v, Tv denotes the
  root-to-v path in T.

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The More Left Than Partial Order on s-to-t paths

• For a nodes s and t, we can define what it means
  for one s-to-t path to be more left than another.
  (Definition omitted comes from Weihe) Implies
  paths dont cross.

75
The More Left Than Partial Order on r-rooted
Trees

• Given two trees TL and TR, both rooted at r, we
  say TL is to the left of TR if, for every node v,
  the r-to-v path TLv is to the left of the
  r-to-v path TRv.

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Right-First Search

• Ripphausen-Lipa, Wagner, Weihe
• Depth-first search where you explore outgoing
  edges from right to left.

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Right-First Search

• Ripphausen-Lipa, Wagner, Weihe
• Depth-first search where you explore outgoing
  edges from right to left.

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Right-First Search

• Ripphausen-Lipa, Wagner, Weihe
• Depth-first search where you explore outgoing
  edges from right to left.
• The right-first search tree T hasthe following
  property
• For any node v, the path Tv is the
  rightmost root-to-v path.

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Single-Source Algorithm

• Start with right-first search tree
• Repeat
• select a leafmost unrelaxed edge
• relax it
• until none exist
• Questions
• Which tree to start with? (Rightmost-search tree)
• How to find a leafmost edge? Use the dual
  dynamic tree.
• How to bound time required? Each relaxation
  requires amortized O(log n) time. Need to bound
  number of relaxations.
• Theorem Each edge is relaxed at most once.
  (Proof)

80
Analysis of Single-Source Algorithm

• Get a sequence T0, T1, T2, of trees.
• To show Each tree Ti1 is to the left of
  previous tree Ti.
• Hence, for each node v, each path Ti1v is to
  the left of the previous path Ti v.

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Analysis of Single-Source Algorithm

• Get a sequence T0, T1, T2, of trees.
• To show Each tree Ti1 is to the left of
  previous tree Ti.
• Hence, for each node v, each path Ti1v is to
  the left of the previous path Ti v.
• Hence these paths use edges entering v in
  clockwise order.
• Use analysis technique outlined at start of talk.

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Invariant

• We say a tree T is right-short if, for every node
  v, Tv is the unique shortest root-to-v path
  that is to the right of Tv.

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Invariant

• We say a tree T is right-short if, for every node
  v, Tv is the unique shortest root-to-v path
  that is to the right of Tv.

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Invariant

• We say a tree T is right-short if, for every node
  v, Tv is the unique shortest root-to-v path
  that is to the right of Tv.
• A right-first search tree is trivially
  right-short.

85
Invariant

• We say a tree T is right-short if, for every node
  v, Tv is the unique shortest root-to-v path
  that is to the right of Tv.
• A right-first search tree is trivially
  right-short.
• Theorem Suppose T is right-short and T is
  obtained from T by relaxing a leafmost unrelaxed
  edge. Then T is right-short and is to the
  left of T.

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• Consider an unrelaxed edge xy.
• Suppose the path Tx ? xy is to the right of
  Ty.

• xy is unrelaxed gt Tx ? xy is shorter than
  Ty.
• Ty is not the shortest path to the right of
  Ty.
• T is not right-short.

87

• Consider an unrelaxed edge xy.
• Suppose the path Tx ? xy is to the right of
  Ty.

• xy is unrelaxed gt Tx ? xy is shorter than
  Ty.
• Ty is not the shortest path to the right of
  Ty.
• T is not right-short.

Lemma 1 If T is right-short and xy is unrelaxed
then Tx ? xy is to the left of Ty.
88

• Theorem Suppose T is right-short and T is
  obtained from T by relaxing a leafmost unrelaxed
  edge xy. Then T is (1) to the left of T and is
  (2) right-short.
• Proof (1) Use Lemma 1 to show that T is to the
  left of T

89

• Theorem Suppose T is right-short and T is
  obtained from T by relaxing a leafmost unrelaxed
  edge xy. Then T is (1) right-short and is (2)
  to the left of T.
• Proof
• Edge xy forms a cycle with the tree.

90

• Theorem Suppose T is right-short and T is
  obtained from T by relaxing a leafmost unrelaxed
  edge xy. Then T is right-short and is to the
  left of T.
• Proof
• Edge xy forms a cycle with the tree.
• Cycle encloses a region R containing no
  unrelaxed edges (since xy wasleafmost unrelaxed
  edge).

91

• Theorem Suppose T is right-short and T is
  obtained from T by relaxing a leafmost unrelaxed
  edge xy. Then T is right-short and is to the
  left of T.
• Proof
• Assume that T is not right-short.

92

• Theorem Suppose T is right-short and T is
  obtained from T by relaxing a leafmost unrelaxed
  edge xy. Then T is right-short and is to the
  left of T.
• Proof
• Assume that T is not right-short.
• Then, for some node v, some path P to theright
  of T v is no longer than T v.

93

• Theorem Suppose T is right-short and T is
  obtained from T by relaxing a leafmost unrelaxed
  edge xy. Then T is right-short and is to the
  left of T.
• Proof
• Assume that T is not right-short.
• Then, for some node v, some path P to theright
  of Tv is no longer than Tv.
• Hence P shorter than Tv.
• But P cant be shorter by going tothe right of
  Tv (would violateright-shortness of T)

94

• Theorem Suppose T is right-short and T is
  obtained from T by relaxing a leafmost unrelaxed
  edge xy. Then T is right-short and is to the
  left of T.
• Proof
• Assume that T is not right-short.
• Then, for some node v, some path P to theright
  of Tv is no longer than Tv.
• Hence P shorter than Tv.
• But P cant be shorter by going tothe right of
  Tv (would violate right-shortness of T)
• And P cant be shorter by using an unrelaxed
  edge inside R (no such edge)

95

• Theorem Suppose T is right-short and T is
  obtained from T by relaxing a leafmost unrelaxed
  edge xy. Then T is right-short and is to the
  left of T.
• Proof
• Assume that T is not right-short.
• Then, for some node v, some path P to theright
  of Tv is no longer than Tv.
• Hence P shorter than Tv.
• But P cant be shorter by going tothe right of
  Tv (would violateright-shortness of T)
• And P cant be shorter by using an unrelaxed
  edge inside R(no such edge)

96

• Theorem Suppose T is right-short and T is
  obtained from T by relaxing a leafmost unrelaxed
  edge xy. Then T is right-short and is to the
  left of T.
• Proof
• Assume that T is not right-short.
• Then, for some node v, some path P to theright
  of Tv is no longer than Tv.
• Hence P shorter than Tv.
• But P cant be shorter by going tothe right of
  Tv (would violateright-shortness of T)
• And P cant be shorter by using an unrelaxed
  edge inside R(no such edge)
• Contradiction. Q.E.D.

97
Multiple-Source Algorithm

• Order the nodes r1,r2,,r10 clockwise around
  graph.
• Add 8-cost edges from r10?r9,, r3?r2,r2?r1
• Start with some tree rooted at r1
• For i 1,2,
• Repeat
• select a leafmost unrelaxed edge
• relax it
• Until no unrelaxed edges remain
• Go from ri-rooted tree to ri1-rooted tree by
  adding edge ri1 ? ri

98
Analysis of Multiple-Source Algorithm

• Go from ri-rooted tree to ri1-rooted tree by
  adding edge ri1?ri
• Must show tree remains right-short.
• Must show same analysis technique applies to
  entire sequence of trees.

99
Multiple-Source Algorithm

• Start with some tree rooted at r1
• For i 1,2,
• Repeat
• find a leafmost unrelaxed edge
• relax it
• Until no unrelaxed edges remain
• Go from ri-rooted tree to ri1-rooted tree by
  adding edge ri1 ? ri
• How to find distances?

100
How to Find Distances (in Multiple Source
Algorithm)

• Start with some tree rooted at r1
• For i 1,2,
• Repeat
• find a leafmost unrelaxed edge
• relax it
• Until no unrelaxed edges remain
• Go from ri-rooted tree to ri1-rooted tree by
  adding edge ri1 ? ri
• Idea use dynamic tree to represent tentative
  shortest-path tree T.
• To relax vw is to cut the current parent edge uv,
  and then join the trees using vw. O(log n) time
• Can query a node v to find sum of costs on
  root-to-v path. O(log n) time
• At the end of iteration I of the for-loop, can
  query for distances from ri.
• Amortized time for total of k queries O(k log n)

101
Conclusion

• Conceptually simple algorithm (details in the
  data structure!)
• Analysis technique might be more generally
  applicable
• Can we compute arbitrary source-to-sink distances
  in O(log n) time per distance?