:, , throughput '

1
???? ??? ?????

• ?? ???????? ????? ?????? ???????? ???? ?????
  ????? ???? ????? ????????

???? ???????? 4 ???? ??? ??? ?????? ??? ?????
????? ??????'?
??? ??????? ?? ????????
??? ??????? ????? ????? ?? ????? ???????? ?????
????? ?? ??? ??? ?? ???? ??? ????? ?????. ?????
??? ??????? ??? ?? ???? ??????? ???? ???????
?????. ???? ???? ????? ????? ?????? ??? ????? ???
???? ?? ??? ????????? ????? ??? ????? ???. ???
????? ????? ??? ?????? ?? ????????? ??? ?????
?????? ????????? ??????. ???????? ?? ????? ??????
?????????? ?"????" ???????? ???.
2
????? ????? ???????

• ????? ?????????? ????? ?? ?? ???
  ????????????????, ????? ???????, throughput
  ????.
• 1. Topology Interference Number ???? ??????? ??
  ???? ????, ??????? ?? ??? ????? ????? ??? ????
  ?????.
• (??? ???? 2 ???? ???)

• 2. ????? ??????? ?? ??? ????? ??????? ????
  ?????? ?????? ??? ??? ????? ???? ???? ?????
  ??????.
• 3. ?????? ???? ???? ?? ??? ????? ???? ????? ??
  ??? ???? ???? ???? ??? ???? ???.

• ???????? ?? ?????? ???? ???????? ?? ???????? ?
  alpha power spanner
• ????? ?? ?????????? ????? ???????? ????? ???????
  ???? ??????? ???? ????? ????? ????? ???? ?-
  Topology Interference Number ????.
• ???? ?????? ?? alpha power spanner ????? ?? ??
  ?????? ??????? ???????? ????? ????, ??- Topology
  interference number ???? ?????? ????? ???? ????
  ???? ?????? ???? ???.

3
t
w
v
r
u
4
Alpha Power Spanner

• ????? alpha power spanner
• ?????? V ???? ??????.
• ???? ????? ??? ??? ????? u,v ????? ???
• u-gtv ?? ??? ?? ??? w ????? ?- V\u,v
  ??????
• ???? d(u,v) ???? ?? ????? ??? u ?- v .

??? u-gtv ???? ???? ?-alpha spanner ?? ??? ??
???? ?? ???? w ????? ????? ??????.
???? 1
5
??????? ???????

• ?????? ?? ????????? ?- C.
• ????? ???? ????? ???? Topology Interference
  Number ????? ????????, ???? ?????? ??? ????
  (????? ???????? ).
• ????? ?? ?????? ????? ??? ?? N ?? (???? ??????
  ????).
• ???? ?? N ?????? 5000 ????? ???? ????? ?? ??? ??
  ?????? ??????? ???? ?????? ????? ?? ??????? ???
  ???? ???? ??????? ???.

• ??????? ???? ???????? ?????? ????? ?????????
  ????? ???
• ????? ?????????? ???? ?????? ?????????? ??????
  ?? ????????? ?-
• ??? ?? ????????? ?????? ???? ????? ??????. ??
  ???? ???? ?? ?????? ??? ?????.

6
????? ????????? ????

• Distributed algorithm for links creation
• Before the algorithm starts we have N random
  points (r,theta) on unit circle and every point
  has unsorted list L of these N points. The goal
  of this algorithm is creation of edges to
  neighbors for every point U.
• for every point U do
• 1 pass over list L and find point V with the
  shortest distance to U.
• 2 () create edge U?V , and remove V from list
  L.
• 3 pass over list L and remove all points that V
  eliminates
• from being neighbors of U
• //for every point W in list L do
• // if dist(U,V)a dist(V,W)a lt dist(U,W)a
• // remove W from list L. ( figure a)
• 4 if list L is not empty return to 1, if
  empty sign that point U has no more edges.
• 5 check if inserted edges are real edges of
  a-spanner
• (()it is possible that redundant edges were
  inserted) (figure b)
• // for every inserted point V in list of
  neighbors do
• // pass over all N points and remove V if exists
  point that
• // eliminates V from being a neighbor of
  U
• // for every point W in list of all points
  do
• // if dist(U,V)a gt dist(U,W)a dist(W,V)a
• // remove V.

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if dist(U,V)a dist(V,W)a lt dist(U,W)a remove
W from list L.
The problem of the redundant edges
U-gtW is a redundant edge because U-gtX-gtW is a
shorter path than U-gtW But we have already
eliminated X from the List L .
8

• ()
• Statement
• If we look for a neighbor V for point U among
  group of points G, it is enough to find the point
  that is closest to U in this group.
• Proof
• Suppose we find a point V that is
  closest to U, and suppose that it cannot be a
  neighbor of U. Hence exists another point W that
  prevents V from being a neighbor of U.
• therefore
• dist(U,W)a dist(W,V)a lt dist(U,V)a
• But according to the assumption V is the closest
  point to U therefore
• dist(U,V) lt dist(U,W)
• dist(U,V)a lt dist(U,W)a
• If so we get
• dist(U,W)a dist(W,V)a lt
  dist(U,V)a lt dist(U,W)a
• dist(W,V)a lt 0
• This can happen only if W and V are the same
  point.
• Contradiction.
• (figure d)

Assumption V is the closest point to U.
It is not possible that dist(U-gtV)a gtdist(
U-gtW)a dist(W-gtV)a
9
???????? ??? ?????? (????)

• () in 3 we pass over list L and remove all
  points that V eliminates from being a neighbor of
  U, but these points that we have eliminated can
  themselves eliminate some other points that we
  will add in the future! Due to this we need to
  recheck all inserted points in 5.
• Pay attention that the number of redundant points
  is very small and is also bounded (we checked it
  in our simulation ) so it does not affect the
  algorithms complexity.

10
?????? ?????????

• Correctness
• The Algorithm is correct because
• 1) For every point U if some edge is supposed to
  be in a-spanner graph we will insert it.
  explanation we remove from the list L points
  that were eliminated by other points, so they can
  not be neighbors of U, and all points that
  were not removed from L after all iterations are
  candidates for being neighbors of U.
• 2)We insert only edges that should be in
  a-spanner and also some redundant edges, and in
  the end we remove all redundant edges, so we get
  the correct a-spanner graph.

11
???????? ??? (????????)

• Complexity
• We checked in our simulation and saw that in the
  average case the number
• of edges for every point is small and can be
  regarded as a constant.
• Therefore ,in the algorithm for every point U,
  after a constant number of
• steps the list L will be empty, and all edges
  will be inserted.
• The Algorithm that every point U runs works as
  follows
• In the average case
  const(O(N)O(1)O(N)O(1))O(N)) O(N)
• 
  
  
• 
  edges part 1 part2 part3 part4
  part5
• In the worst case the point can have all N-1
  edges, we will remove no point in paragraph 3,
  and list L will be empty after N-1 steps.(???? 3)
• In the worst case (N-1)(O(N)O(1)O(N)O(1)O(N
  )) O(N2)
• Hence the algorithm for all N points in the
  circle will work
• In the average case O(N2)
• In the worst case O(N3)

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This is an example of the worst case scenario.
All N-1 points of the Graph have the same
distance to U so they are all neighbors of U . In
contrast in the average case the number of
neighbors is a constant.
???? 3
13
???????? (????)

• In addition we can show that every algorithm for
  building a-spanner is bounded from below by
  O(N2). This is because we need to know for every
  node its distance from every other node in the
  graph.
• Also there exists N2 edges and every algorithm
  will need to check at least all of the edges.
• therefore this operation takes O(N2) actions.

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???? ??????? ???????
Topology Interference Number ???? ??????
??????? ???? ??????? ??? ???? ????? ????. (???
???? 2 ?????? ???? ?????? ?????? ?? ?????).
15
???? ??????? ??????? (????)

• Average Interference Number ???? ???? ???????
  ?? ?? ?????? ???? ???? ???? ?????? ????. ?????
  ???? ?? ??? ?? ??? ?? ???? ?????? ???? ?????? .
  ?????? ???? ?? ???? ?? ????? ???? ??????? ?????
  ?????? ????.

16
???? ??????? ??????? (????)
N runs till 300 Each time we ran our program For
100 times

• Maximum Number of Links (for a Node) ?????
  ???????? ?? ????? ????????
• ????? ?????? ????.(???? ?? ???? ??? ?????? ?????
  ?????? ?? ????? ???????? ????).

17
???? ??????? ??????? (????)
Average Number of Links for a Node ???? ?????
?? ????? ???????? ????? ?? ???? ????. ?????
?????? ?? ???? ?????? ???? ?? ?????? ???? ???????
????? ?????? ????.
18
???? ??????? ??????? (????)
Average redundant edges for a node ???? ?????
?? ????? ??????? ???? ??????? ?????????. ?????
???? ?? ???? ?? counter ????? ?? ???? ??????
???????? ?????? ????? ????? ??? ????? ??? ??.
?????? counter ?? ???? ?? ?????? ??????? ?????
???? ????.
19
???? ??????? ??????? (????)

• Maximum redundant edges per node ???? ???????
  ?? ????? ???????
• ??????? ???? ???? ????? ????.

20
?????? ??????? ?? ??????

• ???? ???? ??
• maximum number of links for a node
• ?????? ????? ?? ??????? ???? ??????? ????? ?????
  ??? ????????
• ??? ?? ????? ?? maximum redundant edges per node
  ??????? ????? ??????? ?????? ???? ?? ??????
• ???? ?????????? ???? ???? ?? ???? ?? ?????? .
• ?????? ????? ???? ????? ????? ?? maximum number
  of links per node
• ???? ?? ??? ???? ???? . ?????? ???? ????? ??????
  ???
• ??? ?? ??? ??? . ??? ?? ???? ?? ?????? ????
  ?????? ????????? ?????? ????? ???? .
• ??? ?????? ??????? ?????? ???? ????? ??? ??????
  ?????? ??????? ????? ??? ??????
• ?????? ??????.

21
?????

• ?????? ???????? ?? ???????? ?-alpha power spanner
  ????? ?? ??????????.
• ?????? ????? ?????????? ???? ????????? ????? ??
  . ??? ?????? ???????? ????? ??? ????
  ????????? ?? .
• ?????? ???????? ????? ????????? ????? ???????
  ????, ?????? ??????? ?????.
• ?????? ?? ?????? ??????? ??????? ??????? matlab
  ???????? ???????. ??????? ?? ?????? ??????
  ??????.

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Thanks, The End