Finding diameter-reducing shortcuts in trees

Davide Bilò, Luciano Gualà, Stefano Leucci, Luca Pepè Sciarria

Abstract: In the k-Diameter-Optimally Augmenting Tree Problem we are given a tree T of n vertices embedded in an unknown metric space. An oracle can report the cost of any edge in constant time, and we want to augment T with k shortcuts to minimize the resulting diameter. When k=1<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">1</mn></math>, O(nlog⁡n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math>-time algorithms exist for paths and trees. We show that o(n2)<math><mi is="true">o</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> queries cannot provide a better than 10/9-approximation for trees when k≥3<math><mi is="true">k</mi><mo is="true">≥</mo><mn is="true">3</mn></math>. For any constant ε>0<math><mi is="true">ε</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></math>, we design a linear-time (1+ε)<math><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mi is="true">ε</mi><mo stretchy="false" is="true">)</mo></math>-approximation algorithm for paths when k=o(log⁡n)<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mi is="true">o</mi><mo stretchy="false" is="true">(</mo><msqrt is="true"><mrow is="true"><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">n</mi></mrow></msqrt><mo stretchy="false" is="true">)</mo></math>, thus establishing a dichotomy between paths and trees for k≥3<math><mi is="true">k</mi><mo is="true">≥</mo><mn is="true">3</mn></math>. Our algorithm employs an ad-hoc data structure, which we also use in a linear-time 4-approximation algorithm for trees, and to compute the diameter of (possibly non-metric) graphs with n+k−1<math><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></math> edges in time O(nklog⁡n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mi is="true">k</mi><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math>.

External IDs:dblp:journals/jcss/BiloGLS25