Comparing Queues and Stacks As Machines for Laying Out Graphs

Abstract

The relative powers of queues and stacks are compared as mechanisms for laying out the edges of a graph. In a k-queue layout, vertices of the graph are placed in some linear order (also called a linear arrangement), and each edge is assigned to exactly one of the k queues, so that the edges assigned to each queue obey a first-in/first-out (FIFO) discipline. As the vertices are scanned left to right, an edge is enqueued on its assigned queue when its left endpoint is encountered and is dequeued from its queue when its right endpoint is encountered. In a k-stack layout, vertices of the graph are placed in some linear order, and each edge is assigned to exactly one of the k stacks so that the edges assigned to each stack obey a last-in/first-out discipline. As the vertices are scanned left to right, an edge is pushed on its assigned stack when its left endpoint is encountered and is popped from its stack when its right endpoint is encountered. The paper has three main results. First, a tradeoff between queuenumber and stacknumber is shown for a fixed linear order of the vertices of G. In particular, for a fixed-order layout of a graph G, \[ {\text{queuenumber }} \times {\text{ stacknumber }} \geq {\text{ cutwidth/valence }}( G ).\] Second, it is shown that every 1-queue graph has a 2-stack layout and that every 1-stack graph has a 2-queue layout. Third, in a surprising display of the power of queues, it is shown that the ternary hypercube requires exponentially more stacks than queues. More precisely, an N-vertex ternary hypercube has a $( 2\log _3 N )$-queue layout but requires $\Omega ( N^{1/9 - \epsilon } )$ stacks, $\epsilon > 0$, in any stack layout. Also, some asymptotic bounds for the queuenumber of bounded-valence graphs are derived.