Developing New Locality Results for the Prüfer Code using a Remarkable Linear-Time Decoding Algorithm

Abstract

The Prüfer Code is a bijection between the $n^{n-2}$ trees on the vertex set $[1,n]$ and the $n^{n-2}$ strings in the set $[1,n]^{n-2}$ (known as Prüfer strings of order $n$). Efficient linear-time algorithms for decoding (i.e., converting string to tree) and encoding (i.e., converting tree to string) are well-known. In this paper, we examine an improved decoding algorithm (due to Cho et al.) that scans the elements of the Prüfer string in reverse order, rather than in the usual forward direction. We show that the algorithm runs in linear time without requiring additional data strutures or sorting routines, and is an 'online' algorithm — every time a new string element is read, the algorithm can correctly output an additional tree edge without any knowledge of the future composition of the string. This new decoding algorithm allows us to derive results concerning the 'locality' properties of the Prüfer Code (i.e., the effect of making small changes to a Prüfer string on the structure of the corresponding tree). First, we show that mutating the $\mu$th element of a Prüfer string (of any order) causes at most $\mu + 1$ edge-changes in the corresponding tree. We also show that randomly mutating the first element of a random Prüfer string of order $n$ causes two edge-changes in the corresponding tree with probability $2(n-3)/n(n-1)$, and one edge-change otherwise. Then, based on computer-aided enumerations, we make three conjectures concerning the locality properties of the Prüfer Code, including a formula for the probability that a random mutation to the $\mu$th element of a random Prüfer string of order $n$ causes exactly one edge-change in the corresponding tree. We show that if this formula is correct, then the probability that a random mutation to a random Prüfer string of order $n$ causes exactly one edge-change in the corresponding tree is asymptotically equal to one-third, as $n$ tends to infinity.

On the Locality of the Prüfer Code

“…In [5], Paulden and Smith conjectured that P (∆ = ℓ > 1 | µ) was on the order of n −1 (conjecture 3 on page 16). We agree with this conjecture, even though we have only proved that P (∆ = ℓ > 1 | µ) is on the order of n −1/3+o (1) .…”

“…In a recent paper [5], Paulden and Smith use combinatorial and numerical methods to develop conjectures about the exact value of P (∆ = ℓ | µ) for ℓ = 1, 2, and about the generic form that P (∆ = ℓ | µ) would take for ℓ > 2. These conjectures, if true, would prove (1.1)- (1.2).…”

“…(1.4) Of course (1.3) implies (1.1), because 1 0 (1 − α) 2 dα = 1/3. In order to prove these results we will need to analyze the following P -string decoding algorithm, which we learned of from [1], [5].…”

Prufer Coding: A Vectorization Method for Undirected Labeled Graph

“…Prufer sequence encoding refers to converting a tree into a character string, and decoding refers to converting a character string into a tree [14] .…”

“…Reference [18] uses simple arrays to improve prufer algorithm, which can improve the time complexity of prufer coding to O(n). Reference [14] studied a decoding algorithm that scanned the prufer sequence in reverse order and proved that the algorithm could run in linear time without the need for additional data structures or sorting processes.…”