Enumerating Distances Using Spanners of Bounded Degree

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Enumerating Distances Using Spanners of Bounded
Degree
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner
Generalizing skip lists A spanner with
logarithmic spanner diameter
Maya Zalcberg 049823446 Ilia Flax
015600052
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Outline
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Enumerating distances using spanners of bounded
  degree
• approximate distance enumeration
• Exact distance enumeration
• Generalizing skip lists A spanner with
  logarithmic spanner diameter

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Introduction
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• The goal
• Enumating the smallest k distances among the
  distances in a group of n points.
• input
• S a set of n points in
• k an integer such that 1 lt k lt

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Introduction
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Output
• a sequence a1,b1ak,bk distinct
  pairs of S is called the k closest pairs, if
• 1.
• 2. the distances ,
  are the k smallest elements in
  multiset

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Introduction
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• There are two solutions available
• using bounded-degree spanner to enumerate the k
  approximate closest pairs in
• Modify the algorithm to obtain the k exact
  closest pairs in

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Approximate distance
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• S a set of n points in , k an integer such
  that , t gt 1 a real number.
• sorted by their distances
• is a sequence of k
  t-approximate closest pairs if

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Approximate distance
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• G (S,E) it a t-spanner for S
• D positive integer. The degree of each point from
  S is less then or equal to D.
• PQ is a priority queue that stores at most k
  pairs.
• Each priority(p,q) in PQ is equal to the length
  of the shortest path in G between p and q at the
  moment.

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Approximate distance
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Dijkstras algorithm
• We use a variant of Dijkstras SingleSource(G,s,R)
  algorithm
• that takes as input an undirected graph G in
  which every edge has a positive weight, a vertex
  s of G, and a real number Rgt0. It returns a set A
  of all vertices v for which

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Algorithm
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• The algorithm uses the t-spanner G to compute a
  sequence of k t-approximate closest pairs in S.
• The idea is to run Dijkstras single-source
  shortest path algorithm simultaneously from all
  point of S.

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Algorithm
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• ApproxDistEnum(G,k)
• Initializing PQ contains the min(k,E) shortest
  edges of E.
• . The priority(p,q) of any such edge p,q is
  equal to the Euclidean distance between p and q.
• Then a sequence of k iterations is carried out.

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Algorithm
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• In each iteration
• The pair p,q with the lowest priority from PQ
  is deleted and reported immediately.
• For each edge q,r (and, symmetrically p,s )
  of G
• If the pair p,r doesnt occur in PQ, then
• priority(p,r)priority(p,q)qr and p,r
  inserted into PQ. If PQ contains k1 pairs, the
  pair with the highest priority is deleted.
• If the pair p,r occures in PQ, then if its
  priority is higher than priority(p,q)qr ,
  then the priorityp,r is set to
  priority(p,q)qr.

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Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner
Correctness

• According to the analysis of Dijkstras
  algorithm, the algorithm returns the k shortest
  path distances in G.
• We show that these k pairs are the t-approximate
  closest pairs in G.

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Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner
Correctness

• Some new definitions
• the length of a shortest path between
  and in the t-spanner G
• is a permutation of 1,2, such that

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Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner
Correctness

• ApproxDistEnum(G,k) reports k pairs of points
  whose final priority values are
• Since G is a t-spanner,
• so by replacing i by we get

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Correctness
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Proof
• 
  

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Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner
Correctness

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Correctness
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

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Correctness
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• For each i, 1ltilt
• The k pairs of points that are reported by the
  algorithm are the k t-approximate closest pairs,
  if

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Correctness
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Proof
• Reminder!
• the length of the shortest path in G between
  and is equal to .
• Else G is a t-spanner, so

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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Initializing
• Building PQ
• Running Dijkstra

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Exact Distance
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• G (S,E) it a t-spanner for point set S, such
  that the maximum degree is D.
• By making two modifications to ApproxDistEnum(G,k)
  we can enumerate the k exact closest pairs.

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Algorithm
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• This algorithm takes as input a t-spanner G
  (S,E) and an integer k.
• It returns a sequence of k exact closest pairs in
  S.

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Algorithm
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• ExactDistEnum(G,k)
• We make two modification to ApproxDistEnum(G,k)
• 1. The priority queue PQ is maintained
  at full size.
• 2. The algorithm does not report pairs as
  before. It keeps the k closest pairs among
  all pairs that were ever inserted into PQ.
• to be continued.

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Algorithm
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• ExactDistEnum(G,k)
• 2. the terminition is when the smallest
  priority in PQ is larger then t times the
  Euclidian-distance of the k-th closest pair
  found so far.
• . At termination, the k closest pairs that have
  been found are reported.

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Algorithm
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Implementation issues
• A second priority queue is maintained that
  contains the same pairs as PQ, and in which the
  priority of and pair p,q is equal to the
  euclidian distance pq.

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Correctness
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Claim
• Let x be the Euclidian distance of the k-th pair
  reported by the algorithm.
• Let p,q be any two distinct points of S such that
  the pair p,q is never inserted into PQ.
• pqgtx

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Correctness
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Proof
• Let r,s a pair that causes us to stop.
• At the termination time priority(r,s)gttx
• Also, at the same time priority(r,s) is equal
  to the length of the shortest path between r and
  s in G.
• At termination, all the shortest paths in G
  shorter then priority(r,s) have been found.

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Correctness
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Proof
• - the length of the shortest path in G
  between p and q.
• priority(r,s).
• Since G is a t-spanner,

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Correctness
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Summary
• For each pair p,q, with , is
  inserted into PQ.
• The distance of the k-th closest pair in S is
  less then or equal to x.
• ExactDistEnum(G,k) enumerates the k closest pairs
  in S.

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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Fundamental claim
• - the k-th smallest Euclidean distance in S.
• p,q the current pair with minimum priority in
  PQ.
• Assume p,q is the first pair for which
• ExactDistEnum(G,k) terminates at the moment when
  it selects p,q

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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Proof
• the length of the shortest path in G between
  p and q
• x the k-th smallest Euclidean distance
• Priority(p,q)
• We have to show that gt tx

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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Proof
• Any pair r,s whose distance in G has already
  occurred as minimal element in PQ.
• r,s distinct points of S such that
• the length of a shortest path in G between r
  and s.

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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Summary
• At p,q selection as a minimal element in PQ,
  all pairs of distinct points of S having
  Euclidean distance at most Wk have been inserted
  into PQ.
• gt tx

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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Summary
• I the number of iterations made by
  ExactDistEnum(G,k)
• Then the running time is
• I is less than or equal to the number of pairs of
  distinct points of S having distance at most
• Now we need to give an upper bound to I

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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Lemma
• - the k-th smallest Euclidean distance in S.
• t a real number, larger then 1.
• M the number of pairs of distinct points of S
  having distance at most

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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Proof
• - d-dimensional grid with cells of side
  lengths . Each cell of this grid has the
  form
• for some integers . a cell is the
  Cartesian product of d intervals, which are
  closed on the left and open on the right.

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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• a cell of is nonempty, if it contains at
  least one point of S.
• g the number of nonempty cells. we number these
  cell arbitrarily .
• - the number of points of S that are
  contained in the i-th nonempty cell.
• , the total number of pairs x,y
  such that x and y are contained in the same cell
  of .

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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• if two points are in the same cell, their
  distance is less .
• w is the k-th smallest distance in S, therefore
  .
• , and an arbitrary cell in ,
• the neighborhood of C in the d-dimensional
  hypercube is defines as followed

To be continued.
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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• the side lengths of are equal to
  , and contains the cell c in its
  center. Neighborhood of C the set of all
  nonempty cells of that overlap .
• For each i, . is the set of all
  indices j, such that the j-th nonempty cell is in
  the neighborhood of the i-th nonempty cell.

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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• p,q two distinct points of s such that
  . p,q are contained in the i-th and j-th
  nonempty cells of .
• if
• but therefore .

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Time Analysis
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• 16. Finally, we can prove the upper bound on M,
  which is the number of distances in S that are
  less then or equal to .

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Summary
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Theorem
• Let S be a set on n points in , let
  be a real number, let be a positive integer,
  and let be a t-spanner for S of degree .
  Given any integer with ,
  algorithm ExactDistEnum(G,k) computes a sequence
  of exact closest pairs in S in

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Generalizing Skip Lists
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• We present a randomized algorithm that constructs
  a sparse spanner whose expected spanner diameter
  is .
• Skip lists this randomized data structure is a
  sparse 1-spanner whose spanner diameter is
  .
• Lets start by presenting some definitions

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Generalizing Skip Lists
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Sparse Spanner a spanner with
  or with a spanner with small weight/size.
• Spanner Diameter The longest shortest path
  between a pair of points in S.
• Next we will present the skip list data
  structure.

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Generalizing Skip Lists
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• S a set on n real numbers.
• Construct a sequence of subsets of S
• and .
• As long as ,construct a random subset
  of as followed
• Each number of has a 50 chance of getting
  picked.
• Set to be the subset of all the picked
  points of .
• Finally, Set and iterate.

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Generalizing Skip Lists
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• - the number of iterations. Then we obtain the
  following sequence
• of subsets of S, where .
• Definition , . The Skip List for
  S1. for each i, , there is a double
  linked list storing the elements of in
  sorted order. Elements of are at level i of
  the skip list.2. for each i, , and
  each , the occurrences of in and
  are connected by pointers.

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Generalizing skip lists
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Example

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Generalizing Skip Lists
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Lemma
• , , produced by the given random
  process, and let M be the total size of the
  corresponding skip list.1. The expected value of
  is . 2. There is a constant C
  that for all large enough real numbers S
  3. M is proportional to , which has
  expected value of . 4. There is a
  constant that for all large enough real
  numbers S

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Building the paths
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• p,q two points in S. we want to build a path
  between p and q.
• Start at p at level 1. walk right until you reach
  an point at level 2.
• If q is reached finished.
• Otherwise, let p1 be the first point encountered
  at level 2.
• Do the same walking left from q at level 1- let
  q1 is the first point encountered at level 2. -
  if p1 is reached finished.
• Recursively build a path between p1 and q1.

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Building the paths
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• This way, we construct two paths.
• The construction stops as soon as the two paths
  meet.
• The result of building a path between 4 and 14 on
  the skip list example is

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Building the paths
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• The expected number of steps made to build a path
  between p and q is
  .
• The excepted number of steps made at level i is
  O(1), and there are levels.

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The spanner in 1-D
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• By flattening the vertices, we can regard the
  skip list as a graph G.
• The expected path length between two points p,q
  is .
• The path the from the example in G is
  4,5,9,12,14.
• The skip list is a 1-spanner for S.
• What is the expected spanner diameter ?

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Summary
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Theorem
• Let S be a set on n real numbers. The skip list
  of S can be regarded as a graph which is a
  1-spanner for S. with high probability, the
  number of edges of this graph is and
  its spanner diameter is .

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Skip list spanner
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• We want to generalize out solution for 1-D to
  2-D.
• S a set of n points in the plane.
• Well use the same random process to obtain the
  following sequence
• of subsets of S, where .

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Skip List 2-D spanner
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Definition
• - an integer. , are
  constructed be the random process.
  For S is defined as follows
• 1. For each, i, , the points of are
  stored in the graph . The points of
  are at level i of the skip list spanner.
• 2. for each i, , and each ,
  the occurrences of in level i and level i-1
  are connected by pointers.

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Recall T graph
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Definition
• Let be an integer, let
  and let S be a set of points in the plane. The
  undirected graph is defined as
  follows
• 1. The vertices of are the points
  of S.
• 2. For each point p of S and for each cone
  C of Ck, such that the translated cone Cp
  contains one or more points of ,
  the graph contains one edge where
  r is a point in , whose
  orthogonal projection onto is closest
  to p

57
Recall T graph
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner
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Skip List 2-D spanner
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• is a graph with vertex set S and
  edge set the union of the edge sets of the graphs
  , .
• Lemma
• an integer, , S a set of n
  points in the plane. The skip list spanner is a
  t-spanner for S with with high probability,
  this graph contains edges and can be
  constructed in time.

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Building a path
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Like in the 1-D case in order to build a path
  between p and q we build two paths. One starting
  at p and the other at q. We stop when the paths
  meet.
• We start at p at level 1 and build a path in the
  graph from p toward q.
• Suppose weve built a path from p to x.
• If x q finished.
• If and x doesnt occur on level 2
• 1. C a cone of such that q in .
• 2. x is the point of such that
  x,x is an edge of .

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Building a path
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Then x is the next point on the path from p
  toward q. set xx.
• We keep extending this path until xq or the
  point x occurs at level 2.
• Assume that x occurs at level 2.
• We start building a path from q to x in
  .
• Assume weve built a path from q to y.
• We stop extending the path if y is one of the
  points on the path from p to x or if y occurs at
  level 2.

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Building a path
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Assume y is a point p on the path from p to x.
  In this case we return the path from p to p
  followed by the reverse of the path from q to p
  and stop.
• Otherwise, x and y are both on level 2 and we use
  the same algorithm to construct a path between
  them.

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Summary
Outline Enumerating distance Using spanners
Approximate distance Presentation
Algorithm Correctness Time
Analysis Exact distance Presentation
Algorithm Correctness Time
Analysis Generalizing skip lists Skip
List Skip List Spanner

• Theorem
• Let be an integer, let ,
  and let be a set of n points in the plane. Then
  the following is true 1. The skip list spanner
  is a t-spanner for
  with high probability it
  contains edges. 2. The skip list
  spanner can be constructed in
  with high probability. 3. with high
  probability, the spanner diameter of the
  skip list spanner is . 4. all these
  bounds are with respect to the coin flips
  that are used to build the skip list
  spanner.

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• Thank You!