Algorithms: Greedy Algorithms

// Greedy Interval Scheduling in Java
//
// PURPOSE
// Earliest-finish-time scheduling selects a maximum-size collection of non-overlapping
// intervals.
//
// METHOD
// Process intervals sorted by finish time. Accept the next interval whenever its start is at
// least the finish of the last accepted interval.
//
// COMPLEXITY
// O(n) scan time and O(1) auxiliary space once sorted; sorting unsorted intervals first
// costs O(n log n).
//
// CONTRACT
// start and finish arrays have n entries; finish is nondecreasing and start[i]<finish[i].
// Half-open intervals allow touching endpoints. Returns selected count.
// 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: 4.
// 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 interval_schedule(int[] start, int[] finish) {
        java.util.Objects.requireNonNull(start, "start");
        java.util.Objects.requireNonNull(finish, "finish");
        int n = start.length;
        alg_require(((n >= 0) && (n <= 1000000)));
        alg_require((start.length == n));
        alg_require((finish.length == n));
        int count = 0;
        int end = 0;
        int i = 0;
        while ((i < n)) {
            if (((count == 0) || (start[i] >= end))) {
                count = Math.addExact(count, 1);
                end = finish[i];
            }
            i = Math.addExact(i, 1);
        }
        return count;
    }

    public static void main(String[] args) {
        int[] start = new int[] {1, 3, 0, 5, 8, 5};
        int[] finish = new int[] {2, 4, 6, 7, 9, 9};
        int result = interval_schedule(start, finish);
        System.out.println(result);
    }
}

"use strict";
// Greedy Interval Scheduling in JavaScript
//
// PURPOSE
// Earliest-finish-time scheduling selects a maximum-size collection of non-overlapping
// intervals.
//
// METHOD
// Process intervals sorted by finish time. Accept the next interval whenever its start is at
// least the finish of the last accepted interval.
//
// COMPLEXITY
// O(n) scan time and O(1) auxiliary space once sorted; sorting unsorted intervals first
// costs O(n log n).
//
// CONTRACT
// start and finish arrays have n entries; finish is nondecreasing and start[i]<finish[i].
// Half-open intervals allow touching endpoints. Returns selected count.
// 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: 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 interval_schedule(start, finish) {
    alg_require(start instanceof Int32Array);
    alg_require(finish instanceof Int32Array);
    let n = start.length;
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require((start.length == n));
    alg_require((finish.length == n));
    let count = 0;
    let end = 0;
    let i = 0;
    while ((i < n)) {
        if (((count == 0) || (start[i] >= end))) {
            count = alg_add(count, 1);
            end = finish[i];
        }
        i = alg_add(i, 1);
    }
    return count;
}

let start = new Int32Array([1, 3, 0, 5, 8, 5]);
let finish = new Int32Array([2, 4, 6, 7, 9, 9]);
let result = interval_schedule(start, finish);
console.log(result);

from bisect import bisect_left

# Greedy Interval Scheduling in Python
#
# PURPOSE
# Earliest-finish-time scheduling selects a maximum-size collection of non-overlapping
# intervals.
#
# METHOD
# Process intervals sorted by finish time. Accept the next interval whenever its start is at
# least the finish of the last accepted interval.
#
# COMPLEXITY
# O(n) scan time and O(1) auxiliary space once sorted; sorting unsorted intervals first
# costs O(n log n).
#
# CONTRACT
# start and finish arrays have n entries; finish is nondecreasing and start[i]<finish[i].
# Half-open intervals allow touching endpoints. Returns selected count.
# 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: 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 interval_schedule(start: list[int], finish: list[int]) -> int:
    alg_require(isinstance(start, (list, tuple)) and len(start) <= 1000000)
    alg_require(isinstance(finish, (list, tuple)) and len(finish) <= 1000000)
    n: int = len(start)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(start) == n)
    alg_require(len(finish) == n)
    count: int = 0
    end: int = 0
    i: int = 0
    while i < n:
        if count == 0 or start[i] >= end:
            count += 1
            end = finish[i]
        i += 1
    return count

start = [1, 3, 0, 5, 8, 5]
finish = [2, 4, 6, 7, 9, 9]
result = interval_schedule(start, finish)
print(result)

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Greedy Interval Scheduling in C
//
// PURPOSE
// Earliest-finish-time scheduling selects a maximum-size collection of non-overlapping
// intervals.
//
// METHOD
// Process intervals sorted by finish time. Accept the next interval whenever its start is at
// least the finish of the last accepted interval.
//
// COMPLEXITY
// O(n) scan time and O(1) auxiliary space once sorted; sorting unsorted intervals first
// costs O(n log n).
//
// CONTRACT
// start and finish arrays have n entries; finish is nondecreasing and start[i]<finish[i].
// Half-open intervals allow touching endpoints. Returns selected count.
// 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: 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 interval_schedule(const int *start, size_t start_length, const int *finish, size_t finish_length) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    if (start_length > 1000000 || (start_length && !start)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (finish_length > 1000000 || (finish_length && !finish)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    int n = (int)start_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)start_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)finish_length == n))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int count = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int end = 0;
    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; }
    while (1) {
        int alg_condition = (i < n);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        int alg_condition_1 = ((count == 0) || (start[i] >= end));
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_1) {
            count = alg_add(count, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            end = finish[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; }
    }
    alg_result = count;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    return alg_result;
}

int main(void) {
    int start[] = {1, 3, 0, 5, 8, 5};
    int finish[] = {2, 4, 6, 7, 9, 9};
    int result = interval_schedule(start, 6, finish, 6);
    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>
// Greedy Interval Scheduling in C++
//
// PURPOSE
// Earliest-finish-time scheduling selects a maximum-size collection of non-overlapping
// intervals.
//
// METHOD
// Process intervals sorted by finish time. Accept the next interval whenever its start is at
// least the finish of the last accepted interval.
//
// COMPLEXITY
// O(n) scan time and O(1) auxiliary space once sorted; sorting unsorted intervals first
// costs O(n log n).
//
// CONTRACT
// start and finish arrays have n entries; finish is nondecreasing and start[i]<finish[i].
// Half-open intervals allow touching endpoints. Returns selected count.
// 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: 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 interval_schedule(const std::vector<int>& start, const std::vector<int>& finish) {
    alg_require(start.size() <= 1000000);
    alg_require(finish.size() <= 1000000);
    int n = (int)start.size();
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((int)start.size() == n));
    alg_require(((int)finish.size() == n));
    int count = 0;
    int end = 0;
    int i = 0;
    while ((i < n)) {
        if (((count == 0) || (start[i] >= end))) {
            count = alg_add(count, 1);
            end = finish[i];
        }
        i = alg_add(i, 1);
    }
    return count;
}

int main(void) {
    std::vector<int> start = {1, 3, 0, 5, 8, 5};
    std::vector<int> finish = {2, 4, 6, 7, 9, 9};
    int result = interval_schedule(start, finish);
    printf("%d\n", result);
    return 0;
}