Algorithms: Range Queries

// Iterative Segment Tree Range Sum in Java
//
// PURPOSE
// A segment tree range query combines the small number of disjoint nodes covering a half-
// open interval.
//
// METHOD
// Move both endpoints to leaves. Include a left endpoint when it is a right child and a
// right endpoint when it is about to cross a left child, then move both up.
//
// COMPLEXITY
// O(log n) time, O(1) auxiliary space.
//
// CONTRACT
// tree has 2*n entries; n>=1, 0<=left<=right<=n. Returns sum in [left,right). Empty range
// returns zero.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// Java 17+: primitive int arrays, Arrays.binarySearch for LIS, and Math exact arithmetic.
// Invalid input throws IllegalArgumentException; arithmetic overflow throws
// ArithmeticException.
//
// EXAMPLE
// Expected return: 5.
// Run: javac AlgorithmExample.java && java AlgorithmExample
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class AlgorithmExample {
    static void alg_require(boolean valid) { if (!valid) throw new IllegalArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int segment_sum(int[] tree, int left, int right) {
        java.util.Objects.requireNonNull(tree, "tree");
        int n = (tree.length / 2);
        alg_require(((n >= 0) && (n <= 1000000)));
        alg_require((tree.length == Math.multiplyExact(2, n)));
        alg_require(((n >= 1) && (left >= 0) && (left <= right) && (right <= n)));
        left = Math.addExact(left, n);
        right = Math.addExact(right, n);
        int total = 0;
        while ((left < right)) {
            if (((left % 2) == 1)) {
                total = Math.addExact(total, tree[left]);
                left = Math.addExact(left, 1);
            }
            if (((right % 2) == 1)) {
                right = Math.subtractExact(right, 1);
                total = Math.addExact(total, tree[right]);
            }
            left = (left / 2);
            right = (right / 2);
        }
        return total;
    }

    public static void main(String[] args) {
        int[] tree = new int[] {0, 10, 3, 7, 1, 2, 3, 4};
        int left = 1;
        int right = 3;
        int result = segment_sum(tree, left, right);
        System.out.println(result);
    }
}

"use strict";
// Iterative Segment Tree Range Sum in JavaScript
//
// PURPOSE
// A segment tree range query combines the small number of disjoint nodes covering a half-
// open interval.
//
// METHOD
// Move both endpoints to leaves. Include a left endpoint when it is a right child and a
// right endpoint when it is about to cross a left child, then move both up.
//
// COMPLEXITY
// O(log n) time, O(1) auxiliary space.
//
// CONTRACT
// tree has 2*n entries; n>=1, 0<=left<=right<=n. Returns sum in [left,right). Empty range
// returns zero.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// Node.js 18+: Int32Array inputs and workspaces provide compact integer storage. Automatic
// lengths and checked arithmetic reject values outside signed 32-bit. Invalid inputs or
// overflow throw RangeError.
//
// EXAMPLE
// Expected return: 5.
// Run: node example.js
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(valid) { if (!valid) throw new RangeError("invalid algorithm input"); }
function alg_checked(v) { if (!Number.isInteger(v) || v < -2147483648 || v > 2147483647) throw new RangeError("32-bit result overflow"); return v; }
function alg_add(a,b) { return alg_checked(a+b); }
function alg_sub(a,b) { return alg_checked(a-b); }
function alg_mul(a,b) { return alg_checked(a*b); }
function alg_size(n) { alg_require(Number.isInteger(n) && n>=0 && n<=4000000); return n; }

function segment_sum(tree, left, right) {
    alg_require(tree instanceof Int32Array);
    alg_require(Number.isInteger(left) && left >= -2147483648 && left <= 2147483647);
    alg_require(Number.isInteger(right) && right >= -2147483648 && right <= 2147483647);
    let n = Math.floor(tree.length / 2);
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require((tree.length == alg_mul(2, n)));
    alg_require(((n >= 1) && (left >= 0) && (left <= right) && (right <= n)));
    left = alg_add(left, n);
    right = alg_add(right, n);
    let total = 0;
    while ((left < right)) {
        if (((left % 2) == 1)) {
            total = alg_add(total, tree[left]);
            left = alg_add(left, 1);
        }
        if (((right % 2) == 1)) {
            right = alg_sub(right, 1);
            total = alg_add(total, tree[right]);
        }
        left = Math.floor(left / 2);
        right = Math.floor(right / 2);
    }
    return total;
}

let tree = new Int32Array([0, 10, 3, 7, 1, 2, 3, 4]);
let left = 1;
let right = 3;
let result = segment_sum(tree, left, right);
console.log(result);

from bisect import bisect_left

# Iterative Segment Tree Range Sum in Python
#
# PURPOSE
# A segment tree range query combines the small number of disjoint nodes covering a half-
# open interval.
#
# METHOD
# Move both endpoints to leaves. Include a left endpoint when it is a right child and a
# right endpoint when it is about to cross a left child, then move both up.
#
# COMPLEXITY
# O(log n) time, O(1) auxiliary space.
#
# CONTRACT
# tree has 2*n entries; n>=1, 0<=left<=right<=n. Returns sum in [left,right). Empty range
# returns zero.
# Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
# counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
# Sorted order, correct tree topology, and caller-owned data-structure invariants are
# preconditions; checking them on every query would change query complexity.
# Scalar and shape checks are included. Value-range checks are included where needed to
# prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
# workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
# uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
# partly updated after arithmetic failure. These are single-threaded, in-memory
# implementations.
#
# LANGUAGE DETAILS
# Python 3.9+: automatic lengths, native lists, arbitrary-precision arithmetic, and
# bisect_left for LIS. Invalid scalar/shape inputs raise ValueError. Integer element types
# and sorted/tree invariants remain caller preconditions.
#
# EXAMPLE
# Expected return: 5.
# Run: python example.py
# Tests for this exact source artifact run before publication. The accompanying source
# archive includes unit cases, runtime results, and source hashes.

def alg_require(condition):
    if not condition:
        raise ValueError("invalid algorithm input; see contract")
def alg_size(n):
    alg_require(0 <= n <= 4000000)
    return n

def segment_sum(tree: list[int], left: int, right: int) -> int:
    alg_require(isinstance(tree, (list, tuple)) and len(tree) <= 1000000)
    alg_require(type(left) is int and -2147483648 <= left <= 2147483647)
    alg_require(type(right) is int and -2147483648 <= right <= 2147483647)
    n: int = len(tree) // 2
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(tree) == 2 * n)
    alg_require(n >= 1 and left >= 0 and (left <= right) and (right <= n))
    left += n
    right += n
    total: int = 0
    while left < right:
        if left % 2 == 1:
            total += tree[left]
            left += 1
        if right % 2 == 1:
            right -= 1
            total += tree[right]
        left = left // 2
        right = right // 2
    return total

tree = [0, 10, 3, 7, 1, 2, 3, 4]
left = 1
right = 3
result = segment_sum(tree, left, right)
print(result)

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Iterative Segment Tree Range Sum in C
//
// PURPOSE
// A segment tree range query combines the small number of disjoint nodes covering a half-
// open interval.
//
// METHOD
// Move both endpoints to leaves. Include a left endpoint when it is a right child and a
// right endpoint when it is about to cross a left child, then move both up.
//
// COMPLEXITY
// O(log n) time, O(1) auxiliary space.
//
// CONTRACT
// tree has 2*n entries; n>=1, 0<=left<=right<=n. Returns sum in [left,right). Empty range
// returns zero.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// C99: checked calloc/free workspace, const input pointers, explicit size_t lengths. Errors
// return INT_MIN with errno=EINVAL, ERANGE, or ENOMEM. Check errno before using a result.
// Input/output arrays must not alias. No variable-length stack arrays.
//
// EXAMPLE
// Expected return: 5.
// Run: gcc -std=c99 -O2 example.c -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static int alg_checked(int64_t v, int *error) {
    if (v < INT_MIN || v > INT_MAX) { *error = 1; return 0; }
    return (int)v;
}
static int alg_add(int a, int b, int *e) { return alg_checked((int64_t)a+b,e); }
static int alg_sub(int a, int b, int *e) { return alg_checked((int64_t)a-b,e); }
static int alg_mul(int a, int b, int *e) { return alg_checked((int64_t)a*b,e); }

int segment_sum(const int *tree, size_t tree_length, int left, int right) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    if (tree_length > 1000000 || (tree_length && !tree)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    int n = ((int)tree_length / 2);
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((n >= 0) && (n <= 1000000)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((int)tree_length == alg_mul(2, n, &alg_error)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((n >= 1) && (left >= 0) && (left <= right) && (right <= n)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    left = alg_add(left, n, &alg_error);
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    right = alg_add(right, n, &alg_error);
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int total = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (left < right);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        int alg_condition_1 = ((left % 2) == 1);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_1) {
            total = alg_add(total, tree[left], &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            left = alg_add(left, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        int alg_condition_2 = ((right % 2) == 1);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_2) {
            right = alg_sub(right, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            total = alg_add(total, tree[right], &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        left = (left / 2);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        right = (right / 2);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    alg_result = total;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    return alg_result;
}

int main(void) {
    int tree[] = {0, 10, 3, 7, 1, 2, 3, 4};
    int left = 1;
    int right = 3;
    int result = segment_sum(tree, 8, left, right);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Iterative Segment Tree Range Sum in C++
//
// PURPOSE
// A segment tree range query combines the small number of disjoint nodes covering a half-
// open interval.
//
// METHOD
// Move both endpoints to leaves. Include a left endpoint when it is a right child and a
// right endpoint when it is about to cross a left child, then move both up.
//
// COMPLEXITY
// O(log n) time, O(1) auxiliary space.
//
// CONTRACT
// tree has 2*n entries; n>=1, 0<=left<=right<=n. Returns sum in [left,right). Empty range
// returns zero.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// C++17: const vector references for inputs, mutable references for outputs, RAII workspace,
// standard lower_bound for LIS. Invalid inputs throw invalid_argument; overflow throws
// overflow_error; allocation failure propagates bad_alloc.
//
// EXAMPLE
// Expected return: 5.
// Run: g++ -std=c++17 -O2 example.cpp -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static void alg_require(bool valid) { if (!valid) throw std::invalid_argument("invalid algorithm input"); }
static int alg_checked(int64_t v) { if (v < INT32_MIN || v > INT32_MAX) throw std::overflow_error("32-bit result overflow"); return (int)v; }
static int alg_add(int a,int b) { return alg_checked((int64_t)a+b); }
static int alg_sub(int a,int b) { return alg_checked((int64_t)a-b); }
static int alg_mul(int a,int b) { return alg_checked((int64_t)a*b); }
static size_t alg_size(int n) { alg_require(n>=0 && n<=4000000); return (size_t)n; }

int segment_sum(const std::vector<int>& tree, int left, int right) {
    alg_require(tree.size() <= 1000000);
    int n = ((int)tree.size() / 2);
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((int)tree.size() == alg_mul(2, n)));
    alg_require(((n >= 1) && (left >= 0) && (left <= right) && (right <= n)));
    left = alg_add(left, n);
    right = alg_add(right, n);
    int total = 0;
    while ((left < right)) {
        if (((left % 2) == 1)) {
            total = alg_add(total, tree[left]);
            left = alg_add(left, 1);
        }
        if (((right % 2) == 1)) {
            right = alg_sub(right, 1);
            total = alg_add(total, tree[right]);
        }
        left = (left / 2);
        right = (right / 2);
    }
    return total;
}

int main(void) {
    std::vector<int> tree = {0, 10, 3, 7, 1, 2, 3, 4};
    int left = 1;
    int right = 3;
    int result = segment_sum(tree, left, right);
    printf("%d\n", result);
    return 0;
}