zig/lib/std / treap.zig

const std = @import("std.zig");
const assert = std.debug.assert;
const testing = std.testing;
const Order = std.math.Order;

pub fn Treap(comptime Key: type, comptime compareFn: anytype) type {
    return struct {
        const Self = @This();

        // Allow for compareFn to be fn (anytype, anytype) anytype
        // which allows the convenient use of std.math.order.
        fn compare(a: Key, b: Key) Order {
            return compareFn(a, b);
        }

        root: ?*Node = null,
        prng: Prng = .{},

        /// A customized pseudo random number generator for the treap.
        /// This just helps reducing the memory size of the treap itself
        /// as std.Random.DefaultPrng requires larger state (while producing better entropy for randomness to be fair).
        const Prng = struct {
            xorshift: usize = 0,

            fn random(self: *Prng, seed: usize) usize {
                // Lazily seed the prng state
                if (self.xorshift == 0) {
                    self.xorshift = seed;
                }

                // Since we're using usize, decide the shifts by the integer's bit width.
                const shifts = switch (@bitSizeOf(usize)) {
                    64 => .{ 13, 7, 17 },
                    32 => .{ 13, 17, 5 },
                    16 => .{ 7, 9, 8 },
                    else => @compileError("platform not supported"),
                };

                self.xorshift ^= self.xorshift >> shifts[0];
                self.xorshift ^= self.xorshift << shifts[1];
                self.xorshift ^= self.xorshift >> shifts[2];

                assert(self.xorshift != 0);
                return self.xorshift;
            }
        };

        /// A Node represents an item or point in the treap with a uniquely associated key.
        pub const Node = struct {
            key: Key,
            priority: usize,
            parent: ?*Node,
            children: [2]?*Node,

            pub fn next(node: *Node) ?*Node {
                return nextOnDirection(node, 1);
            }
            pub fn prev(node: *Node) ?*Node {
                return nextOnDirection(node, 0);
            }
        };

        fn extremeInSubtreeOnDirection(node: *Node, direction: u1) *Node {
            var cur = node;
            while (cur.children[direction]) |next| cur = next;
            return cur;
        }

        fn nextOnDirection(node: *Node, direction: u1) ?*Node {
            if (node.children[direction]) |child| {
                return extremeInSubtreeOnDirection(child, direction ^ 1);
            }
            var cur = node;
            // Traversing upward until we find `parent` to `cur` is NOT on
            // `direction`, or equivalently, `cur` to `parent` IS on
            // `direction` thus `parent` is the next.
            while (true) {
                if (cur.parent) |parent| {
                    // If `parent -> node` is NOT on `direction`, then
                    // `node -> parent` IS on `direction`
                    if (parent.children[direction] != cur) return parent;
                    cur = parent;
                } else {
                    return null;
                }
            }
        }

        /// Returns the smallest Node by key in the treap if there is one.
        /// Use `getEntryForExisting()` to replace/remove this Node from the treap.
        pub fn getMin(self: Self) ?*Node {
            if (self.root) |root| return extremeInSubtreeOnDirection(root, 0);
            return null;
        }

        /// Returns the largest Node by key in the treap if there is one.
        /// Use `getEntryForExisting()` to replace/remove this Node from the treap.
        pub fn getMax(self: Self) ?*Node {
            if (self.root) |root| return extremeInSubtreeOnDirection(root, 1);
            return null;
        }

        /// Lookup the Entry for the given key in the treap.
        /// The Entry act's as a slot in the treap to insert/replace/remove the node associated with the key.
        pub fn getEntryFor(self: *Self, key: Key) Entry {
            var parent: ?*Node = undefined;
            const node = self.find(key, &parent);

            return Entry{
                .key = key,
                .treap = self,
                .node = node,
                .context = .{ .inserted_under = parent },
            };
        }

        /// Get an entry for a Node that currently exists in the treap.
        /// It is undefined behavior if the Node is not currently inserted in the treap.
        /// The Entry act's as a slot in the treap to insert/replace/remove the node associated with the key.
        pub fn getEntryForExisting(self: *Self, node: *Node) Entry {
            assert(node.priority != 0);

            return Entry{
                .key = node.key,
                .treap = self,
                .node = node,
                .context = .{ .inserted_under = node.parent },
            };
        }

        /// An Entry represents a slot in the treap associated with a given key.
        pub const Entry = struct {
            /// The associated key for this entry.
            key: Key,
            /// A reference to the treap this entry is apart of.
            treap: *Self,
            /// The current node at this entry.
            node: ?*Node,
            /// The current state of the entry.
            context: union(enum) {
                /// A find() was called for this entry and the position in the treap is known.
                inserted_under: ?*Node,
                /// The entry's node was removed from the treap and a lookup must occur again for modification.
                removed,
            },

            /// Update's the Node at this Entry in the treap with the new node (null for deleting). `new_node`
            /// can have `undefind` content because the value will be initialized internally.
            pub fn set(self: *Entry, new_node: ?*Node) void {
                // Update the entry's node reference after updating the treap below.
                defer self.node = new_node;

                if (self.node) |old| {
                    if (new_node) |new| {
                        self.treap.replace(old, new);
                        return;
                    }

                    self.treap.remove(old);
                    self.context = .removed;
                    return;
                }

                if (new_node) |new| {
                    // A previous treap.remove() could have rebalanced the nodes
                    // so when inserting after a removal, we have to re-lookup the parent again.
                    // This lookup shouldn't find a node because we're yet to insert it..
                    var parent: ?*Node = undefined;
                    switch (self.context) {
                        .inserted_under => |p| parent = p,
                        .removed => assert(self.treap.find(self.key, &parent) == null),
                    }

                    self.treap.insert(self.key, parent, new);
                    self.context = .{ .inserted_under = parent };
                }
            }
        };

        fn find(self: Self, key: Key, parent_ref: *?*Node) ?*Node {
            var node = self.root;
            parent_ref.* = null;

            // basic binary search while tracking the parent.
            while (node) |current| {
                const order = compare(key, current.key);
                if (order == .eq) break;

                parent_ref.* = current;
                node = current.children[@intFromBool(order == .gt)];
            }

            return node;
        }

        fn insert(self: *Self, key: Key, parent: ?*Node, node: *Node) void {
            // generate a random priority & prepare the node to be inserted into the tree
            node.key = key;
            node.priority = self.prng.random(@intFromPtr(node));
            node.parent = parent;
            node.children = [_]?*Node{ null, null };

            // point the parent at the new node
            const link = if (parent) |p| &p.children[@intFromBool(compare(key, p.key) == .gt)] else &self.root;
            assert(link.* == null);
            link.* = node;

            // rotate the node up into the tree to balance it according to its priority
            while (node.parent) |p| {
                if (p.priority <= node.priority) break;

                const is_right = p.children[1] == node;
                assert(p.children[@intFromBool(is_right)] == node);

                const rotate_right = !is_right;
                self.rotate(p, rotate_right);
            }
        }

        fn replace(self: *Self, old: *Node, new: *Node) void {
            // copy over the values from the old node
            new.key = old.key;
            new.priority = old.priority;
            new.parent = old.parent;
            new.children = old.children;

            // point the parent at the new node
            const link = if (old.parent) |p| &p.children[@intFromBool(p.children[1] == old)] else &self.root;
            assert(link.* == old);
            link.* = new;

            // point the children's parent at the new node
            for (old.children) |child_node| {
                const child = child_node orelse continue;
                assert(child.parent == old);
                child.parent = new;
            }
        }

        fn remove(self: *Self, node: *Node) void {
            // rotate the node down to be a leaf of the tree for removal, respecting priorities.
            while (node.children[0] orelse node.children[1]) |_| {
                self.rotate(node, rotate_right: {
                    const right = node.children[1] orelse break :rotate_right true;
                    const left = node.children[0] orelse break :rotate_right false;
                    break :rotate_right (left.priority < right.priority);
                });
            }

            // node is a now a leaf; remove by nulling out the parent's reference to it.
            const link = if (node.parent) |p| &p.children[@intFromBool(p.children[1] == node)] else &self.root;
            assert(link.* == node);
            link.* = null;

            // clean up after ourselves
            node.priority = 0;
            node.parent = null;
            node.children = [_]?*Node{ null, null };
        }

        fn rotate(self: *Self, node: *Node, right: bool) void {
            // if right, converts the following:
            //      parent -> (node (target YY adjacent) XX)
            //      parent -> (target YY (node adjacent XX))
            //
            // if left (!right), converts the following:
            //      parent -> (node (target YY adjacent) XX)
            //      parent -> (target YY (node adjacent XX))
            const parent = node.parent;
            const target = node.children[@intFromBool(!right)] orelse unreachable;
            const adjacent = target.children[@intFromBool(right)];

            // rotate the children
            target.children[@intFromBool(right)] = node;
            node.children[@intFromBool(!right)] = adjacent;

            // rotate the parents
            node.parent = target;
            target.parent = parent;
            if (adjacent) |adj| adj.parent = node;

            // fix the parent link
            const link = if (parent) |p| &p.children[@intFromBool(p.children[1] == node)] else &self.root;
            assert(link.* == node);
            link.* = target;
        }

        /// Usage example:
        ///   var iter = treap.inorderIterator();
        ///   while (iter.next()) |node| {
        ///     ...
        ///   }
        pub const InorderIterator = struct {
            current: ?*Node,

            pub fn next(it: *InorderIterator) ?*Node {
                const current = it.current;
                it.current = if (current) |cur|
                    cur.next()
                else
                    null;
                return current;
            }
        };

        pub fn inorderIterator(self: *Self) InorderIterator {
            return .{ .current = self.getMin() };
        }
    };
}

// For iterating a slice in a random order
// https://lemire.me/blog/2017/09/18/visiting-all-values-in-an-array-exactly-once-in-random-order/
fn SliceIterRandomOrder(comptime T: type) type {
    return struct {
        rng: std.Random,
        slice: []T,
        index: usize = undefined,
        offset: usize = undefined,
        co_prime: usize,

        const Self = @This();

        pub fn init(slice: []T, rng: std.Random) Self {
            return Self{
                .rng = rng,
                .slice = slice,
                .co_prime = blk: {
                    if (slice.len == 0) break :blk 0;
                    var prime = slice.len / 2;
                    while (prime < slice.len) : (prime += 1) {
                        var gcd = [_]usize{ prime, slice.len };
                        while (gcd[1] != 0) {
                            const temp = gcd;
                            gcd = [_]usize{ temp[1], temp[0] % temp[1] };
                        }
                        if (gcd[0] == 1) break;
                    }
                    break :blk prime;
                },
            };
        }

        pub fn reset(self: *Self) void {
            self.index = 0;
            self.offset = self.rng.int(usize);
        }

        pub fn next(self: *Self) ?*T {
            if (self.index >= self.slice.len) return null;
            defer self.index += 1;
            return &self.slice[((self.index *% self.co_prime) +% self.offset) % self.slice.len];
        }
    };
}

const TestTreap = Treap(u64, std.math.order);
const TestNode = TestTreap.Node;

test "insert, find, replace, remove" {
    var treap = TestTreap{};
    var nodes: [10]TestNode = undefined;

    var prng = std.Random.DefaultPrng.init(0xdeadbeef);
    var iter = SliceIterRandomOrder(TestNode).init(&nodes, prng.random());

    // insert check
    iter.reset();
    while (iter.next()) |node| {
        const key = prng.random().int(u64);

        // make sure the current entry is empty.
        var entry = treap.getEntryFor(key);
        try testing.expectEqual(entry.key, key);
        try testing.expectEqual(entry.node, null);

        // insert the entry and make sure the fields are correct.
        entry.set(node);
        try testing.expectEqual(node.key, key);
        try testing.expectEqual(entry.key, key);
        try testing.expectEqual(entry.node, node);
    }

    // find check
    iter.reset();
    while (iter.next()) |node| {
        const key = node.key;

        // find the entry by-key and by-node after having been inserted.
        const entry = treap.getEntryFor(node.key);
        try testing.expectEqual(entry.key, key);
        try testing.expectEqual(entry.node, node);
        try testing.expectEqual(entry.node, treap.getEntryForExisting(node).node);
    }

    // in-order iterator check
    {
        var it = treap.inorderIterator();
        var last_key: u64 = 0;
        while (it.next()) |node| {
            try std.testing.expect(node.key >= last_key);
            last_key = node.key;
        }
    }

    // replace check
    iter.reset();
    while (iter.next()) |node| {
        const key = node.key;

        // find the entry by node since we already know it exists
        var entry = treap.getEntryForExisting(node);
        try testing.expectEqual(entry.key, key);
        try testing.expectEqual(entry.node, node);

        var stub_node: TestNode = undefined;

        // replace the node with a stub_node and ensure future finds point to the stub_node.
        entry.set(&stub_node);
        try testing.expectEqual(entry.node, &stub_node);
        try testing.expectEqual(entry.node, treap.getEntryFor(key).node);
        try testing.expectEqual(entry.node, treap.getEntryForExisting(&stub_node).node);

        // replace the stub_node back to the node and ensure future finds point to the old node.
        entry.set(node);
        try testing.expectEqual(entry.node, node);
        try testing.expectEqual(entry.node, treap.getEntryFor(key).node);
        try testing.expectEqual(entry.node, treap.getEntryForExisting(node).node);
    }

    // remove check
    iter.reset();
    while (iter.next()) |node| {
        const key = node.key;

        // find the entry by node since we already know it exists
        var entry = treap.getEntryForExisting(node);
        try testing.expectEqual(entry.key, key);
        try testing.expectEqual(entry.node, node);

        // remove the node at the entry and ensure future finds point to it being removed.
        entry.set(null);
        try testing.expectEqual(entry.node, null);
        try testing.expectEqual(entry.node, treap.getEntryFor(key).node);

        // insert the node back and ensure future finds point to the inserted node
        entry.set(node);
        try testing.expectEqual(entry.node, node);
        try testing.expectEqual(entry.node, treap.getEntryFor(key).node);
        try testing.expectEqual(entry.node, treap.getEntryForExisting(node).node);

        // remove the node again and make sure it was cleared after the insert
        entry.set(null);
        try testing.expectEqual(entry.node, null);
        try testing.expectEqual(entry.node, treap.getEntryFor(key).node);
    }
}

test "inorderIterator" {
    var treap = TestTreap{};
    var nodes: [10]TestNode = undefined;

    // Build the tree.
    var i: usize = 0;
    while (i < 10) : (i += 1) {
        const key = @as(u64, i);
        var entry = treap.getEntryFor(key);
        entry.set(&nodes[i]);
    }

    // Test the iterator.
    var iter = treap.inorderIterator();
    i = 0;
    while (iter.next()) |node| {
        const key = @as(u64, i);
        try testing.expectEqual(key, node.key);
        i += 1;
    }
}

test "getMin, getMax, simple" {
    var treap = TestTreap{};
    var nodes: [3]TestNode = undefined;

    try testing.expectEqual(null, treap.getMin());
    try testing.expectEqual(null, treap.getMax());
    { // nodes[1]
        var entry = treap.getEntryFor(1);
        entry.set(&nodes[1]);
        try testing.expectEqual(&nodes[1], treap.getMin());
        try testing.expectEqual(&nodes[1], treap.getMax());
    }
    { // nodes[0]
        var entry = treap.getEntryFor(0);
        entry.set(&nodes[0]);
        try testing.expectEqual(&nodes[0], treap.getMin());
        try testing.expectEqual(&nodes[1], treap.getMax());
    }
    { // nodes[2]
        var entry = treap.getEntryFor(2);
        entry.set(&nodes[2]);
        try testing.expectEqual(&nodes[0], treap.getMin());
        try testing.expectEqual(&nodes[2], treap.getMax());
    }
}

test "getMin, getMax, random" {
    var nodes: [100]TestNode = undefined;
    var prng = std.Random.DefaultPrng.init(0xdeadbeef);
    var iter = SliceIterRandomOrder(TestNode).init(&nodes, prng.random());

    var treap = TestTreap{};
    var min: u64 = std.math.maxInt(u64);
    var max: u64 = 0;

    try testing.expectEqual(null, treap.getMin());
    try testing.expectEqual(null, treap.getMax());

    // Insert and check min/max after each insertion.
    iter.reset();
    while (iter.next()) |node| {
        const key = prng.random().int(u64);

        // Insert into `treap`.
        var entry = treap.getEntryFor(key);
        entry.set(node);

        if (key < min) min = key;
        if (key > max) max = key;

        const min_node = treap.getMin().?;
        try std.testing.expectEqual(null, min_node.prev());
        try std.testing.expectEqual(min, min_node.key);

        const max_node = treap.getMax().?;
        try std.testing.expectEqual(null, max_node.next());
        try std.testing.expectEqual(max, max_node.key);
    }
}

test "node.{prev(),next()} with sequential insertion and deletion" {
    // Insert order: 50, 0, 1, 2, ..., 49, 51, 52, ..., 99.
    // Delete order: 0, 1, 2, ..., 49, 51, 52, ..., 99.
    // Check 50's neighbors.
    var treap = TestTreap{};
    var nodes: [100]TestNode = undefined;
    {
        var entry = treap.getEntryFor(50);
        entry.set(&nodes[50]);
        try testing.expectEqual(50, nodes[50].key);
        try testing.expectEqual(null, nodes[50].prev());
        try testing.expectEqual(null, nodes[50].next());
    }
    // Insert others.
    var i: usize = 0;
    while (i < 50) : (i += 1) {
        const key = @as(u64, i);
        const node = &nodes[i];
        var entry = treap.getEntryFor(key);
        entry.set(node);
        try testing.expectEqual(key, node.key);
        try testing.expectEqual(node, nodes[50].prev());
        try testing.expectEqual(null, nodes[50].next());
    }
    i = 51;
    while (i < 100) : (i += 1) {
        const key = @as(u64, i);
        const node = &nodes[i];
        var entry = treap.getEntryFor(key);
        entry.set(node);
        try testing.expectEqual(key, node.key);
        try testing.expectEqual(&nodes[49], nodes[50].prev());
        try testing.expectEqual(&nodes[51], nodes[50].next());
    }
    // Remove others.
    i = 0;
    while (i < 49) : (i += 1) {
        const key = @as(u64, i);
        var entry = treap.getEntryFor(key);
        entry.set(null);
        try testing.expectEqual(&nodes[49], nodes[50].prev());
        try testing.expectEqual(&nodes[51], nodes[50].next());
    }
    { // i = 49.
        const key = @as(u64, i);
        var entry = treap.getEntryFor(key);
        entry.set(null);
        try testing.expectEqual(null, nodes[50].prev());
        try testing.expectEqual(&nodes[51], nodes[50].next());
    }
    i = 51;
    while (i < 99) : (i += 1) {
        const key = @as(u64, i);
        var entry = treap.getEntryFor(key);
        entry.set(null);
        try testing.expectEqual(null, nodes[50].prev());
        try testing.expectEqual(&nodes[i + 1], nodes[50].next());
    }
    { // i = 99.
        const key = @as(u64, i);
        var entry = treap.getEntryFor(key);
        entry.set(null);
        try testing.expectEqual(null, nodes[50].prev());
        try testing.expectEqual(null, nodes[50].next());
    }
}

fn findFirstGreaterOrEqual(array: []u64, value: u64) usize {
    var i: usize = 0;
    while (i < array.len and array[i] < value) i += 1;
    return i;
}

fn testOrderedArrayAndTreapConsistency(array: []u64, treap: *TestTreap) !void {
    var i: usize = 0;
    while (i < array.len) : (i += 1) {
        const value = array[i];

        const entry = treap.getEntryFor(value);
        try testing.expect(entry.node != null);
        const node = entry.node.?;
        try testing.expectEqual(value, node.key);

        if (i == 0) {
            try testing.expectEqual(node.prev(), null);
        } else {
            try testing.expectEqual(node.prev(), treap.getEntryFor(array[i - 1]).node);
        }
        if (i + 1 == array.len) {
            try testing.expectEqual(node.next(), null);
        } else {
            try testing.expectEqual(node.next(), treap.getEntryFor(array[i + 1]).node);
        }
    }
}

test "node.{prev(),next()} with random data" {
    var nodes: [100]TestNode = undefined;
    var prng = std.Random.DefaultPrng.init(0xdeadbeef);
    var iter = SliceIterRandomOrder(TestNode).init(&nodes, prng.random());

    var treap = TestTreap{};
    // A slow, stupid but correct reference. Ordered.
    var golden = std.ArrayList(u64).init(std.testing.allocator);
    defer golden.deinit();

    // Insert.
    iter.reset();
    while (iter.next()) |node| {
        const key = prng.random().int(u64);

        // Insert into `golden`.
        const i = findFirstGreaterOrEqual(golden.items, key);
        // Ensure not found. If found: `prng`'s fault.
        try testing.expect(i == golden.items.len or golden.items[i] > key);
        try golden.insert(i, key);

        // Insert into `treap`.
        var entry = treap.getEntryFor(key);
        entry.set(node);

        try testOrderedArrayAndTreapConsistency(golden.items, &treap);
    }

    // Delete.
    iter.reset();
    while (iter.next()) |node| {
        const key = node.key;

        // Delete from `golden`.
        const i = findFirstGreaterOrEqual(golden.items, key);
        try testing.expect(i < golden.items.len);
        _ = golden.orderedRemove(i);

        // Delete from `treap`.
        var entry = treap.getEntryFor(key);
        try testing.expect(entry.node != null);
        entry.set(null);

        try testOrderedArrayAndTreapConsistency(golden.items, &treap);
    }
}