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Deterministic Multi-Channel Information Exchange. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box .: A A A A A A A A A A. n:= # nodes. Problem :. n:= # nodes k:= # information. Problem :. Have information. ?. Disseminate to all!.
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Deterministic Multi-Channel Information Exchange TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA
n:= # nodes Problem:
n:= # nodes k:= # information Problem: Have information ? Disseminate to all!
Problem: ? Disseminate to all!
Problem: ? Disseminate to all!
n:= # nodes Problem: 1 5 2 3 Unique IDs 1…n 4 n
Problem: ? Disseminate to all! Easy: O(n) Faster?
I can: send / receive reacheachnode
I can: ? send / receive reacheachnode
I can: send / receive reacheachnode nocollisiondetection
I can: send / receive reacheachnode nocollisiondetection 101 Mhz 117 Mhz 132 Mhz … switchchannels synchronus
I can: complexity computation: free radio: time 1 send / receive reacheachnode nocollisiondetection switchchannels synchronus
n:= # nodes k:= # information k
n:= # nodes k:= # information k k
n:= # nodes k:= # information k k Optimal
n:= # nodes k:= # information k k Optimal ????
n:= # nodes k:= # information [HPSW11] - Channels needed for time O(k):
n:= # nodes k:= # information [HPSW11] - Channels needed for time O(k): This paper:
n:= # nodes k:= # information randomized [HPSW11] - Channels needed for time O(k): This paper:
n:= # nodes k:= # information randomized [HPSW11] - Channels needed for time O(k): deterministic This paper:
n:= # nodes k:= # information randomized [HPSW11] - Channels needed for time O(k): deterministic This paper: Optimal? Optimal? Optimal? Optimal?
n:= # nodes k:= # information randomized [HPSW11] - Channels needed for time O(k): deterministic This paper: k
Main ingredient: Specially taylored graphs.
Main ingredient: Specially taylored graphs. (Inspired by use of lossless expanders in [CK08])
Main ingredient: Specially taylored graphs. (Inspired by use of lossless expanders in [CK08]) Topology: Still single hop. Graphs used to select channel.
Bipartite : node IDs new names 1 5 2 6 3 7 4
Bipartite : node IDs new names 1 5 2 6 3 7 4
Matching Graphs: • Nodes in V have degee
Matching Graphs: • Nodes in V have degee • Fixed order of edges
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most k there is at least nodes in X have a unique i-neighbor.
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most k there is at least nodes in X have a unique i-neighbor. 1 5 2 6 i 3 have unique i-neighbor 7 i 4
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most k there is at least nodes in X have a unique i-neighbor. 1 5 2 6 i 3 have unique i-neighbor 7 i 4
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most k there is at least nodes in X have a unique i-neighbor. 1 1 2 5 2 2 1 6 2 3 1 7 2 4 1
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most k there is at least nodes in X have a unique i-neighbor. 1 1 2 5 2 2 1 6 2 3 1 X 7 2 4 1
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most k there is at least nodes in X have a unique i-neighbor. 1 1 5 2 1 6 3 1 X 7 4 1
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most k there is at least nodes in X have a unique i-neighbor. 1 1 5 2 1 6 3 1 X 7 4 BAD 1
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most k there is at least nodes in X have a unique i-neighbor. 1 1 2 5 2 2 1 6 2 3 1 X 7 2 4 1
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most k there is at least nodes in X have a unique i-neighbor. 1 2 5 2 2 6 2 3 X 7 2 4
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most k there is at least nodes in X have a unique i-neighbor. 1 2 5 2 2 6 2 3 X 7 2 GOOD 4
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most k there is at least nodes in X have a unique i-neighbor. 1 1 X 5 2 1 6 3 1 7 4 1
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most k there is at least nodes in X have a unique i-neighbor. 1 1 X 5 2 1 6 3 1 7 4 1
Matching Graphs: • Nodes in V have degee • Fixed order of edges • For any of size at most K there is at least nodes in X have a unique i-neighbor. 1 1 2 5 2 2 1 exist if 6 2 3 1 7 2 4 1
What are these graphs good for? Renaming
What are these graphs good for? Renaming 1 1 2 5 2 2 1 6 2 3 1 7 2 4 1
What are these graphs good for? Renaming • To each of the k «reporters» we can assing a new unique name in |W| in time O( using |W| channels. 1 1 2 5 2 2 1 6 2 3 1 7 2 4 1
What is renaming good for? Assignment of reporters to channels!
What is renaming good for? Assignment of reporters to channels! Example: k < log n
What is renaming good for? Assignment of reporters to channels! Example: k < log n Original names n