Question

Design an algorithm that, when given an arrangement of the digits \(0,1,2,3,4,5,6,7\), 8,9 , rearranges the digits so that the new arrangement represents the next larger value that can be represented by these digits (or reports that no such rearrangement exists if no rearrangement produces a larger value). Thus 5647382901 would produce 5647382910 .

Short answer

Next larger value: 5647382910.

Step-by-step solution

Identify the longest non-increasing suffix

Find the longest non-increasing suffix in the given number. This is done by scanning the number from right to left until you find a digit that is smaller than the digit immediately to its right.

Choose the pivot

The digit immediately preceding the longest non-increasing suffix is called the pivot. This pivot is the number we need to swap to obtain the next permutation.

Find the rightmost successor of the pivot

Find the smallest digit on the right of the pivot that is larger than the pivot itself. This involves scanning the non-increasing suffix from right to left.

Swap the pivot with its rightmost successor

Exchange the pivot with the rightmost successor that was identified in the previous step. This swap will make a larger permutation than the original.

Reverse the suffix

Reverse the suffix starting right after the original position of the pivot. This ensures that the new number constructed is the smallest possible next permutation.

Key concepts

Permutations

Permutations are a fundamental concept in mathematics, especially in the field of combinatorics. When we talk about permutations, we're referring to the different ways in which a set of objects can be arranged or ordered. The key idea is that the order of arrangement matters, which is different from combinations where the order does not matter. In our exercise, we are dealing with permutations of a sequence of digits.
Given a set of 10 digits from 0 to 9, there's a whole array of permutations that can be formed. The goal can be to find a particular permutation with specific properties, like the smallest or largest order, or the next possible order in an increasing series. This involves transforming one particular permutation into another using a step-by-step process, which is critical in understanding algorithms that handle permutations, such as the one described in our example below.

Digit Arrangement

Arranging digits to form different numbers is a practical application of permutations. To specifically address this, we think about changing the order of digits in a number, aiming to achieve different values.
In the exercise, the digits need to be rearranged to form the next largest number possible. The concept of digit arrangement shares close ties with permutations but adds a numerical constraint—the new arrangement must result in the next larger value.
The key steps involve understanding which mismatches in the current sequence can lead to a higher order when corrected. This affects both the identification of critical digits and the strategic placement of those digits to fulfill the desired outcome governed by numerical properties.

Step-by-Step Algorithms

A step-by-step algorithm is a detailed and systematic procedure used to solve a problem efficiently. This particular exercise outlines a simple yet effective algorithm to find the next permutation of a sequence of numbers.
The algorithm begins by identifying the longest non-increasing suffix. This means scanning from the rightmost end of the number to find where the higher digits stop decreasing.
Identifying a pivot, the number that needs to be changed, is crucial. The pivot effectively marks the change point in creating a larger sequence.
The next steps involve finding the right successor—searching for the smallest larger number after the pivot. Then, swapping these elements, followed by reversing the suffix, steps essential for ensuring the newly formed number is the immediate next permutation. Breaking down these steps simplifies the process and highlights the systematic nature of algorithms in solving such problems.

Numerical Algorithms

Numerical algorithms are designed to solve problems that are expressed in numerical terms. They often involve operations like sorting, rearranging, or calculating results efficiently.
In our permutation exercise, the algorithm leverages numerical properties to compute the next rearrangement effectively.
Key numerical techniques include identifying the critical pivot and finding the successor in a controlled manner. By reversing the suffix, the algorithm ensures that the rearranged number is not only a legitimate permutation but also the next consecutive larger order.
These techniques are deeply embedded in numerical algorithms, demonstrating how efficiently a machine can perform calculations that mimic logical human decision-making aimed at achieving specific goals.