Algorithms: Range Queries

"use strict";
// Iterative Segment Tree Build in JavaScript
//
// PURPOSE
// An iterative sum segment tree stores leaves in the second half of a flat array and parents
// in the first half.
//
// METHOD
// Copy values into tree[n+i], then compute every internal node from its two children in
// descending index order.
//
// COMPLEXITY
// O(n) time and O(1) auxiliary space beyond caller-owned tree.
//
// CONTRACT
// n>=1, tree length is 2*n. Build once and reuse for point assignments and half-open range
// queries. Sums must fit.
// 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: 0.
// Expected tree after call: [0, 10, 3, 7, 1, 2, 3, 4].
// 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_build(a, tree) {
    alg_require(a instanceof Int32Array);
    alg_require(tree instanceof Int32Array);
    let n = a.length;
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require((a.length == n));
    alg_require((tree.length == alg_mul(2, n)));
    alg_require((n >= 1));
    let i = 0;
    tree[0] = 0;
    while ((i < n)) {
        tree[alg_add(n, i)] = a[i];
        i = alg_add(i, 1);
    }
    i = alg_sub(n, 1);
    while ((i > 0)) {
        tree[i] = alg_add(tree[alg_mul(2, i)], tree[alg_add(alg_mul(2, i), 1)]);
        i = alg_sub(i, 1);
    }
    return 0;
}

let a = new Int32Array([1, 2, 3, 4]);
let tree = new Int32Array([0, 0, 0, 0, 0, 0, 0, 0]);
let result = segment_build(a, tree);
console.log(result);
// Expected tree: [0, 10, 3, 7, 1, 2, 3, 4]
console.log(Array.from(tree));

from bisect import bisect_left

# Iterative Segment Tree Build in Python
#
# PURPOSE
# An iterative sum segment tree stores leaves in the second half of a flat array and parents
# in the first half.
#
# METHOD
# Copy values into tree[n+i], then compute every internal node from its two children in
# descending index order.
#
# COMPLEXITY
# O(n) time and O(1) auxiliary space beyond caller-owned tree.
#
# CONTRACT
# n>=1, tree length is 2*n. Build once and reuse for point assignments and half-open range
# queries. Sums must fit.
# 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: 0.
# Expected tree after call: [0, 10, 3, 7, 1, 2, 3, 4].
# 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_build(a: list[int], tree: list[int]) -> int:
    alg_require(isinstance(a, (list, tuple)) and len(a) <= 1000000)
    alg_require(isinstance(tree, list) and len(tree) <= 1000000)
    n: int = len(a)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(a) == n)
    alg_require(len(tree) == 2 * n)
    alg_require(n >= 1)
    i: int = 0
    tree[0] = 0
    while i < n:
        tree[n + i] = a[i]
        i += 1
    i = n - 1
    while i > 0:
        tree[i] = tree[2 * i] + tree[2 * i + 1]
        i -= 1
    return 0

a = [1, 2, 3, 4]
tree = [0, 0, 0, 0, 0, 0, 0, 0]
result = segment_build(a, tree)
print(result)
# Expected tree: [0, 10, 3, 7, 1, 2, 3, 4]
print(tree)

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Iterative Segment Tree Build in C
//
// PURPOSE
// An iterative sum segment tree stores leaves in the second half of a flat array and parents
// in the first half.
//
// METHOD
// Copy values into tree[n+i], then compute every internal node from its two children in
// descending index order.
//
// COMPLEXITY
// O(n) time and O(1) auxiliary space beyond caller-owned tree.
//
// CONTRACT
// n>=1, tree length is 2*n. Build once and reuse for point assignments and half-open range
// queries. Sums must fit.
// 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: 0.
// Expected tree after call: [0, 10, 3, 7, 1, 2, 3, 4].
// 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_build(const int *a, size_t a_length, int *tree, size_t tree_length) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    if (a_length > 1000000 || (a_length && !a)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (tree_length > 1000000 || (tree_length && !tree)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    int n = (int)a_length;
    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)a_length == n))) { 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))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int i = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    tree[0] = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (i < n);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        tree[alg_add(n, i, &alg_error)] = a[i];
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        i = alg_add(i, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    i = alg_sub(n, 1, &alg_error);
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (i > 0);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        tree[i] = alg_add(tree[alg_mul(2, i, &alg_error)], tree[alg_add(alg_mul(2, i, &alg_error), 1, &alg_error)], &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        i = alg_sub(i, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    alg_result = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    return alg_result;
}

int main(void) {
    int a[] = {1, 2, 3, 4};
    int tree[] = {0, 0, 0, 0, 0, 0, 0, 0};
    int result = segment_build(a, 4, tree, 8);
    printf("%d\n", result);
    // Expected tree: [0, 10, 3, 7, 1, 2, 3, 4]
    for (int alg_print = 0; alg_print < 8; alg_print++) printf("%d ", tree[alg_print]);
    printf("\n");
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Iterative Segment Tree Build in C++
//
// PURPOSE
// An iterative sum segment tree stores leaves in the second half of a flat array and parents
// in the first half.
//
// METHOD
// Copy values into tree[n+i], then compute every internal node from its two children in
// descending index order.
//
// COMPLEXITY
// O(n) time and O(1) auxiliary space beyond caller-owned tree.
//
// CONTRACT
// n>=1, tree length is 2*n. Build once and reuse for point assignments and half-open range
// queries. Sums must fit.
// 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: 0.
// Expected tree after call: [0, 10, 3, 7, 1, 2, 3, 4].
// 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_build(const std::vector<int>& a, std::vector<int>& tree) {
    alg_require(a.size() <= 1000000);
    alg_require(tree.size() <= 1000000);
    int n = (int)a.size();
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((int)a.size() == n));
    alg_require(((int)tree.size() == alg_mul(2, n)));
    alg_require((n >= 1));
    int i = 0;
    tree[0] = 0;
    while ((i < n)) {
        tree[alg_add(n, i)] = a[i];
        i = alg_add(i, 1);
    }
    i = alg_sub(n, 1);
    while ((i > 0)) {
        tree[i] = alg_add(tree[alg_mul(2, i)], tree[alg_add(alg_mul(2, i), 1)]);
        i = alg_sub(i, 1);
    }
    return 0;
}

int main(void) {
    std::vector<int> a = {1, 2, 3, 4};
    std::vector<int> tree = {0, 0, 0, 0, 0, 0, 0, 0};
    int result = segment_build(a, tree);
    printf("%d\n", result);
    // Expected tree: [0, 10, 3, 7, 1, 2, 3, 4]
    for (int alg_print = 0; alg_print < 8; alg_print++) printf("%d ", tree[alg_print]);
    printf("\n");
    return 0;
}

using System;
// Iterative Segment Tree Build in C#
//
// PURPOSE
// An iterative sum segment tree stores leaves in the second half of a flat array and parents
// in the first half.
//
// METHOD
// Copy values into tree[n+i], then compute every internal node from its two children in
// descending index order.
//
// COMPLEXITY
// O(n) time and O(1) auxiliary space beyond caller-owned tree.
//
// CONTRACT
// n>=1, tree length is 2*n. Build once and reuse for point assignments and half-open range
// queries. Sums must fit.
// 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# / .NET 10: ReadOnlySpan<int> inputs, Span<int> outputs, primitive workspace arrays,
// Array.BinarySearch for LIS, and checked integer arithmetic. Invalid arguments and overflow
// throw exceptions.
//
// EXAMPLE
// Expected return: 0.
// Expected tree after call: [0, 10, 3, 7, 1, 2, 3, 4].
// Run: Use Program.cs in a net10.0 console project; dotnet run.
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

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

    static int segment_build(ReadOnlySpan<int> a, Span<int> tree) {
        checked {
            int n = a.Length;
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require((a.Length == n));
            alg_require((tree.Length == (2 * n)));
            alg_require((n >= 1));
            int i = 0;
            tree[0] = 0;
            while ((i < n)) {
                tree[(n + i)] = a[i];
                i = (i + 1);
            }
            i = (n - 1);
            while ((i > 0)) {
                tree[i] = (tree[(2 * i)] + tree[((2 * i) + 1)]);
                i = (i - 1);
            }
            return 0;
        }
    }

    static void Main() {
        int[] a = new int[] {1, 2, 3, 4};
        int[] tree = new int[] {0, 0, 0, 0, 0, 0, 0, 0};
        int result = segment_build(a, tree);
        Console.WriteLine(result);
        // Expected tree: [0, 10, 3, 7, 1, 2, 3, 4]
        Console.WriteLine("[" + string.Join(", ", tree) + "]");
    }
}