Summary: On-line and Approximation Algorithms Fall Semester, 2011/12
Exercise 1: Nov 23, 2011
Lecturer: Prof. Yossi Azar
Write short but full and accurate answers. Each solution should appear on a separate page and each of its parts
should not exceed a page.
1. Suppose we have a set of requests over the real continuous time. Each request arrives at an arbitrary time
(which is a real number). An algorithm may provide a service point at any time (a real number - not
necessarily immediately after the request). The response time of a request is the time from its arrival until
the next service point. The goal is to minimize the number of service points plus the total response time of
all requests. Note that a service point serves all waiting requests simultaneously (i.e. it serves all requests
that arrive after the previous service point).
(a) Design a 2 competitive deterministic algorithm.
(b) Show a lower bound of 2 (more precisely 2- for any positive ) even if an additive constant is allowed.
Hint: create a request immediately after any service point.
(c) How would your answers to (a) and (b) change if each service point can serve at most k requests for
some fixed k > 1.
2. Suppose you need to find an unknown point (x0, y0) in the Euclidean plane. You start at the origin and you
can move in any continues curve in the plane. You find your point when you reach a point (x0, y) for any y
or a point (x, y0) for any x (you do not need to reach (x0, y0)). Your goal is to minimize the curve's length.
Algorithms are deterministic.