Team Alice

Overall Objectives
Scientific Foundations
Application Domains
New Results
Contracts and Grants with Industry
Other Grants and Activities

Section: New Results

Algorithms and combinatorics

Participant : Laurent Alonso.

Bounds for Cops and Robber Pursuit

We prove that the robber can evade (that is, stay at least unit distance from) at least Im3 ${\#8970 n/5.889\#8971 }$ cops patroling an n×n continuous square region, that a robber can always evade a single cop patroling a square with side length 4 or larger, and that a single cop on patrol can always capture the robber in a square with side length smaller than Im4 ${2.189\#8943 }$ (with E.M. Reingold, submitted to J. of Computational Geometry Theory and Applications).

Average-Case Lower Bounds for the Plurality Problem

Given a set of n elements, each of which is colored one of c$ \ge$2 colors, we have to determine an element of the plurality (most frequently occurring) color by pairwise equal/unequal color comparisons of elements. We derive lower bounds for the expected number of color comparisons when the cn colorings are equally probable. We prove a general lower bound of Im5 ${\mfrac c3n-O{(\sqrt n)}}$ for c$ \ge$2 ; we prove the stronger particular bounds of Im6 ${\mfrac 76n-O{(\sqrt n)}}$ for c = 3 , Im7 ${\mfrac 5435n-O{(\sqrt n)}}$ for c = 4 , Im8 ${\mfrac 607315n-O{(\sqrt n)}}$ for c = 5 , Im9 ${\mfrac 1592693n-O{(\sqrt n)}}$ for c = 6 , Im10 ${\mfrac 79853003n-O{(\sqrt n)}}$ for c = 7 , and Im11 ${\mfrac 194026435n-O{(\sqrt n)}}$ for c = 8 (with E.M. Reigold, [11] ).


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