```
<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
    <title>Find Most Ancestral Elements with Progress Feedback</title>
    <style>
        body {
            font-family: Arial, sans-serif;
            margin: 20px;
            position: relative; /* To position the target rectangle absolutely within the body */
        }

        /* Button Styles */
        .buttons {
            margin-bottom: 20px;
        }

        button {
            padding: 10px 20px;
            margin-right: 10px;
            font-size: 16px;
            cursor: pointer;
        }

        /* Progress Bar Styles */
        #progress-container {
            width: 100%;
            background-color: #f3f3f3;
            border: 1px solid #ccc;
            border-radius: 5px;
            overflow: hidden;
            margin-bottom: 10px;
            height: 25px;
        }

        #progress-bar {
            width: 0%;
            height: 100%;
            background-color: #4caf50;
            transition: width 0.2s ease;
        }

        #progress-text {
            display: block;
            text-align: right;
            margin-top: -25px;
            margin-bottom: 20px;
            font-weight: bold;
        }

        /* Elements Container */
        #elements-container {
            width: 800px; /* Define width */
            height: 600px; /* Define height */
            display: block; /* Change from flex to block for absolute positioning */
            max-height: 600px;
            overflow: auto;
            border: 1px solid #ccc;
            padding: 10px;
            position: relative; /* To position nested elements if needed */
        }

        /* Generated Elements */
        .generated-element {
            width: 50px;
            height: 50px;
            background-color: #add8e6;
            border: 1px solid #000;
            box-sizing: border-box;
        }

        /* Results */
        #results {
            margin-top: 20px;
        }

        #results ul {
            list-style-type: none;
            padding-left: 0;
        }

        #results li {
            padding: 5px 0;
        }

        /* Target Rectangle Styles */
        #target-rect {
            position: fixed;
            border: 2px solid red;
            pointer-events: none; /* Allows clicks to pass through the rectangle */
            z-index: 1000; /* Ensure it's on top of other elements */
            display: none; /* Hidden by default */
        }

        /* Input Sections */
        .input-section {
            margin-bottom: 20px;
        }

        .input-section label {
            display: inline-block;
            width: 120px;
            font-weight: bold;
        }

        .input-section input {
            padding: 5px;
            width: 100px;
            margin-right: 10px;
            box-sizing: border-box;
        }

        .error-message {
            color: red;
            margin-top: 5px;
            display: block;
        }
    </style>
</head>
<body>

    <h1>Find Most Ancestral Elements with Progress Feedback</h1>

    <!-- Generation Parameters Section -->
    <div class="input-section">
        <h2>Generation Parameters</h2>
        <label for="generate-count">Number of Elements:</label>
        <input type="number" id="generate-count" value="200" min="1">
        <label for="generate-max-depth">Maximum Depth:</label>
        <input type="number" id="generate-max-depth" value="2" min="0">
        <button id="generate-btn">Generate Elements</button>
        <span id="generate-errors" class="error-message"></span>
    </div>

    <!-- Target Rectangle Definition Section -->
    <div class="input-section">
        <h2>Define Target Rectangle</h2>
        <label for="rect-x">X-coordinate:</label>
        <input type="number" id="rect-x" value="0" min="0">
        <label for="rect-y">Y-coordinate:</label>
        <input type="number" id="rect-y" value="0" min="0">
        <label for="rect-width">Width:</label>
        <input type="number" id="rect-width" value="100" min="1">
        <label for="rect-height">Height:</label>
        <input type="number" id="rect-height" value="100" min="1">
        <button id="update-rect-btn">Update Rect</button>
        <button id="create-target-btn">Create Random Rect</button>
        <span id="rect-errors" class="error-message"></span>
    </div>

    <!-- Search Button -->
    <div class="buttons">
        <button id="search-btn" disabled>Start Search</button>
    </div>

    <!-- Progress Bar -->
    <div id="progress-container">
        <div id="progress-bar"></div>
    </div>
    <span id="progress-text">0%</span>

    <h2>Generated Elements</h2>
    <div id="elements-container"></div>

    <!-- Target Rectangle -->
    <div id="target-rect"></div>

    <h2>Search Results</h2>
    <div id="results">
        <ul id="results-list"></ul>
    </div>

    <script>
        // Optimized findMostAncestralElements function using closure for caching
        const findMostAncestralElements = (function () {
            // Cached helper functions
            function isMatch(rect, target, z) {
                // Calculate the coordinates of the intersection rectangle
                const xLeft = Math.max(rect.left, target.x);
                const yTop = Math.max(rect.top, target.y);
                const xRight = Math.min(rect.left + rect.width, target.x + target.width);
                const yBottom = Math.min(rect.top + rect.height, target.y + target.height);

                // Compute the width and height of the overlapping area
                const overlapWidth = xRight - xLeft;
                const overlapHeight = yBottom - yTop;

                // If there is no overlap, the overlapping area is 0
                if (overlapWidth <= 0 || overlapHeight <= 0) {
                    return false; // No part of rect is inside target
                }

                // Calculate areas
                const overlapArea = overlapWidth * overlapHeight;
                const rectArea = rect.width * rect.height;

                // Determine if the overlapping area meets the required ratio
                return overlapArea >= (1 - z) * rectArea;
            }

            function isOverlap(rect, target) {
                const { left, top, width: rectWidth, height: rectHeight } = rect;
                const { x, y, width: targetWidth, height: targetHeight } = target;

                return (
                    left < x + targetWidth &&
                    left + rectWidth > x &&
                    top < y + targetHeight &&
                    top + rectHeight > y
                );
            }

            function createProcessBatchAsync(queue, target, z, exhaustive, iterationBatch, onProgress, head, results) {
                return function processBatchAsync() {
                    return new Promise((resolve) => {
                        let processed = 0;

                        while (head.value < queue.length && processed < iterationBatch) {
                            const elem = queue[head.value++];
                            const rect = elem.getBoundingClientRect();

                            if (isMatch(rect, target, z)) {
                                results.add(elem);
                                console.log(elem);
                            } else if (exhaustive || isOverlap(rect, target)) {
                                const children = elem.children;
                                for (let i = 0, len = children.length; i < len; i++) {
                                    queue.push(children[i]);
                                }
                            }

                            processed++;
                        }

                        if (onProgress) {
                            onProgress({
                                processedElements: head.value,
                                totalQueued: queue.length,
                                uniqueResults: results.size,
                            });
                        }

                        resolve();
                    });
                };
            }

            /**
             * The optimized findMostAncestralElements function.
             *
             * @param {number} x - The X-coordinate of the target rectangle (relative to the viewport).
             * @param {number} y - The Y-coordinate of the target rectangle (relative to the viewport).
             * @param {number} width - The width of the target rectangle.
             * @param {number} height - The height of the target rectangle.
             * @param {number} errorRatio - The allowed error ratio (0 to 1) for matching.
             * @param {Object} [options] - Optional settings.
             * @param {boolean} [options.exhaustive=true] - If true, performs an exhaustive search.
             * @param {number} [options.asyncIterations] - Number of iterations per asynchronous batch.
             *                                             If null, undefined, or <= 0, executes synchronously.
             * @param {Function} [options.onProgress] - Callback function to receive progress updates.
             * @returns {Promise<Element[]>|Element[]} A promise that resolves to an array of the most ancestral DOM elements matching the criteria, or the array directly if not asynchronous.
             */
            return function findMostAncestralElements(
                x,
                y,
                width,
                height,
                errorRatio,
                options = {}
            ) {
                const {
                    exhaustive = true,
                    asyncIterations, // Renamed from 'async' to 'asyncIterations'
                    onProgress,
                } = options;

                // Determine if the function should execute asynchronously
                const isAsync = typeof asyncIterations === 'number' && asyncIterations > 0;
                const iterationBatch = isAsync ? asyncIterations : null;

                // Clamp errorRatio between 0 and 1
                const z = Math.max(0, Math.min(errorRatio, 1));

                const target = { x, y, width, height };
                const results = new Set();
                const queue = [document.body];
                const head = { value: 0 }; // Using an object to allow mutation inside closure

                if (isAsync) {
                    const processBatch = createProcessBatchAsync(
                        queue,
                        target,
                        z,
                        exhaustive,
                        iterationBatch,
                        onProgress,
                        head,
                        results
                    );

                    return new Promise((resolve) => {
                        function iterate() {
                            processBatch().then(() => {
                                if (head.value < queue.length) {
                                    setTimeout(iterate, 0);
                                } else {
                                    resolve(Array.from(results));
                                }
                            });
                        }
                        iterate();
                    });
                } else {
                    // Synchronous Processing
                    while (head.value < queue.length) {
                        const elem = queue[head.value++];
                        const rect = elem.getBoundingClientRect();

                        if (isMatch(rect, target, z)) {
                            results.add(elem);
                            continue; // Skip enqueuing children to maintain ancestry
                        }

                        if (exhaustive || isOverlap(rect, target)) {
                            const children = elem.children;
                            for (let i = 0, len = children.length; i < len; i++) {
                                queue.push(children[i]);
                            }
                        }

                        if (onProgress) {
                            onProgress({
                                processedElements: head.value,
                                totalQueued: queue.length,
                                uniqueResults: results.size,
                            });
                        }
                    }
                    return Array.from(results);
                }
            };
        })();

        // Function to update the progress bar and percentage text
        function updateProgressBar(progress) {
            const { processedElements, totalQueued } = progress;
            const progressBar = document.getElementById('progress-bar');
            const progressText = document.getElementById('progress-text');

            // Estimate percentage based on processed vs total queued
            const percentage = totalQueued > 0 ? Math.min((processedElements / (processedElements + totalQueued)) * 100, 100) : 100;

            progressBar.style.width = `${percentage}%`;
            progressText.textContent = `${percentage.toFixed(2)}%`;
        }

        // Corrected Function to Generate Elements in a Grid Without Overlapping
        function generateElements(count = 200, maxDepth = 2) {
            const container = document.getElementById('elements-container');
            container.innerHTML = ''; // Clear previous elements if any

            const ELEMENT_SIZE = 50; // px
            const GAP = 10; // px

            // Get container dimensions
            const containerRect = container.getBoundingClientRect();
            const containerWidth = containerRect.width - GAP; // Adjust for padding and gaps
            const containerHeight = containerRect.height - GAP;

            const columns = Math.floor((containerWidth) / (ELEMENT_SIZE + GAP));
            const rows = Math.floor((containerHeight) / (ELEMENT_SIZE + GAP));

            let currentColumn = 0;
            let currentRow = 0;

            // Function to get position based on row and column
            function getPosition(row, column) {
                const left = column * (ELEMENT_SIZE + GAP);
                const top = row * (ELEMENT_SIZE + GAP);
                return { left, top };
            }

            // Function to create nested elements with controlled branching
            function createNestedElements(parent, currentDepth) {
                if (currentDepth > maxDepth) return;

                // Define a fixed number of child elements per parent to prevent exponential growth
                const CHILDREN_PER_PARENT = 2; // Fixed branching factor

                for (let i = 0; i < CHILDREN_PER_PARENT; i++) {
                    const div = document.createElement('div');
                    div.classList.add('generated-element');

                    // Position children within the parent with a smaller gap
                    const childGap = 5; // px
                    const childLeft = (i % 2) * (ELEMENT_SIZE + childGap);
                    const childTop = Math.floor(i / 2) * (ELEMENT_SIZE + childGap);

                    // Apply inline styles for absolute positioning
                    div.style.position = 'absolute';
                    div.style.left = `${childLeft}px`;
                    div.style.top = `${childTop}px`;

                    parent.appendChild(div);

                    // Randomly decide to nest further with controlled depth
                    if (Math.random() > 0.5) { // 50% chance to nest for demonstration
                        createNestedElements(div, currentDepth + 1);
                    }
                }
            }

            // Generate top-level elements in grid
            for (let i = 0; i < count; i++) {
                if (currentRow >= rows) {
                    console.warn('Reached maximum rows. Some elements may not be visible.');
                    break;
                }

                const div = document.createElement('div');
                div.classList.add('generated-element');

                const row = currentRow;
                const column = currentColumn;
                const { left, top } = getPosition(row, column);

                // Apply inline styles for absolute positioning
                div.style.position = 'absolute';
                div.style.left = `${left}px`;
                div.style.top = `${top}px`;

                container.appendChild(div);

                // Randomly decide to nest further with controlled branching
                if (Math.random() > 0.7) { // 30% chance to nest
                    createNestedElements(div, 1);
                }

                // Update row and column
                currentColumn++;
                if (currentColumn >= columns) {
                    currentColumn = 0;
                    currentRow++;
                }
            }
        }

        // Function to display search results
        function displayResults(elements) {
            const resultsList = document.getElementById('results-list');
            resultsList.innerHTML = ''; // Clear previous results

            if (elements.length === 0) {
                resultsList.innerHTML = '<li>No matching elements found.</li>';
                return;
            }

            elements.forEach((elem, index) => {
                const li = document.createElement('li');
                li.textContent = `Element ${index + 1}: ${elem.tagName}${elem.id ? `#${elem.id}` : ''}${elem.className ? `.${elem.className}` : ''}`;
                resultsList.appendChild(li);
            });
            window.results = Array.from(elements);
        }

        // Function to draw or update the target rectangle
        function drawTargetRect(targetRect) {
            const targetDiv = document.getElementById('target-rect');
            // Set the position and size based on targetRect
            targetDiv.style.left = `${targetRect.x}px`;
            targetDiv.style.top = `${targetRect.y}px`;
            targetDiv.style.width = `${targetRect.width}px`;
            targetDiv.style.height = `${targetRect.height}px`;
            // Display the rectangle
            targetDiv.style.display = 'block';
        }

        // Function to create a random target rectangle
        function createRandomTargetRect() {
            const targetDiv = document.getElementById('target-rect');

            // Define target rectangle dimensions
            const width = 100; // px
            const height = 100; // px

            // Calculate random positions within the container

            const maxX = window.innerWidth;
            const maxY = window.innerHeight;

            const x = Math.random() * maxX;
            const y = Math.random() * maxY;

            const targetRect = { x: Math.round(x), y: Math.round(y), width, height };

            drawTargetRect(targetRect);
            updateRectInputFields(targetRect); // Update input fields with new values
            return targetRect;
        }

        // Function to update the rectangle input fields with given values
        function updateRectInputFields(rect) {
            document.getElementById('rect-x').value = rect.x;
            document.getElementById('rect-y').value = rect.y;
            document.getElementById('rect-width').value = rect.width;
            document.getElementById('rect-height').value = rect.height;
        }

        // Event listeners for buttons
        document.getElementById('generate-btn').addEventListener('click', () => {
            const generateBtn = document.getElementById('generate-btn');
            const createTargetBtn = document.getElementById('create-target-btn');
            const searchBtn = document.getElementById('search-btn');
            const progressBar = document.getElementById('progress-bar');
            const progressText = document.getElementById('progress-text');
            const resultsList = document.getElementById('results-list');
            const generateErrors = document.getElementById('generate-errors');

            // Clear previous error messages
            generateErrors.textContent = '';

            // Get input values
            const countInput = document.getElementById('generate-count').value;
            const maxDepthInput = document.getElementById('generate-max-depth').value;

            let hasError = false;
            let count = parseInt(countInput, 10);
            let maxDepth = parseInt(maxDepthInput, 10);
            let errorMessage = '';

            // Validate count
            if (isNaN(count) || count <= 0) {
                errorMessage += 'Number of Elements must be a positive integer. ';
                hasError = true;
            }

            // Validate maxDepth
            if (isNaN(maxDepth) || maxDepth < 0) {
                errorMessage += 'Maximum Depth must be a non-negative integer.';
                hasError = true;
            }

            if (hasError) {
                generateErrors.textContent = errorMessage;
                return;
            }

            // Disable buttons during generation
            generateBtn.disabled = true;
            createTargetBtn.disabled = true;
            searchBtn.disabled = true;
            resultsList.innerHTML = '';
            progressBar.style.width = '0%';
            progressText.textContent = '0%';

            // Hide the target rectangle when generating new elements
            const targetDiv = document.getElementById('target-rect');
            targetDiv.style.display = 'none';

            // Generate elements
            setTimeout(() => { // Use setTimeout to allow UI to update
                generateElements(count, maxDepth); // Generate elements with user-specified count and maxDepth
                generateBtn.disabled = false;
                createTargetBtn.disabled = false;
                searchBtn.disabled = false;
            }, 100); // Slight delay to ensure buttons get disabled first
        });

        document.getElementById('create-target-btn').addEventListener('click', () => {
            const targetRect = createRandomTargetRect();
            // Optionally, you can notify the user
            // alert('Random Target Rectangle Created!');
        });

        document.getElementById('update-rect-btn').addEventListener('click', () => {
            const rectErrors = document.getElementById('rect-errors');
            rectErrors.textContent = ''; // Clear previous error messages

            // Get input values
            const xInput = document.getElementById('rect-x').value;
            const yInput = document.getElementById('rect-y').value;
            const widthInput = document.getElementById('rect-width').value;
            const heightInput = document.getElementById('rect-height').value;

            let hasError = false;
            let x = parseFloat(xInput);
            let y = parseFloat(yInput);
            let width = parseFloat(widthInput);
            let height = parseFloat(heightInput);
            let errorMessage = '';

            // Validate x
            if (isNaN(x) || x < 0) {
                errorMessage += 'X-coordinate must be a non-negative number. ';
                hasError = true;
            }

            // Validate y
            if (isNaN(y) || y < 0) {
                errorMessage += 'Y-coordinate must be a non-negative number. ';
                hasError = true;
            }

            // Validate width
            if (isNaN(width) || width <= 0) {
                errorMessage += 'Width must be a positive number. ';
                hasError = true;
            }

            // Validate height
            if (isNaN(height) || height <= 0) {
                errorMessage += 'Height must be a positive number.';
                hasError = true;
            }

            if (hasError) {
                rectErrors.textContent = errorMessage;
                return;
            }

            const targetRect = { x, y, width, height };
            drawTargetRect(targetRect);
            console.log('User Defined Target Rectangle:', targetRect);
            // Optionally, notify the user
        });

        document.getElementById('search-btn').addEventListener('click', () => {
            const searchBtn = document.getElementById('search-btn');
            const resultsList = document.getElementById('results-list');

            // Clear previous results
            resultsList.innerHTML = '';
            // Reset progress bar
            const progressBar = document.getElementById('progress-bar');
            const progressText = document.getElementById('progress-text');
            progressBar.style.width = '0%';
            progressText.textContent = '0%';

            // Disable buttons during search
            const generateBtn = document.getElementById('generate-btn');
            const createTargetBtn = document.getElementById('create-target-btn');
            const updateRectBtn = document.getElementById('update-rect-btn');
            generateBtn.disabled = true;
            createTargetBtn.disabled = true;
            updateRectBtn.disabled = true;
            searchBtn.disabled = true;

            let targetRect = getCurrentTargetRect();

            // If no target rect exists, create one
            if (!targetRect) {
                targetRect = createRandomTargetRect();
            }

            // Log the target rectangle
            console.log('Using Target Rectangle:', targetRect);

            // Start search
            findMostAncestralElements(
                targetRect.x,
                targetRect.y,
                targetRect.width,
                targetRect.height,
                0.5, // errorRatio
                {
                    asyncIterations: 100, // Process 100 elements per batch
                    onProgress: (progress) => {
                        updateProgressBar(progress);
                    }
                }
            ).then((matchingElements) => {
                displayResults(matchingElements);
                alert(`Search complete! Found ${matchingElements.length} matching elements.`);
                generateBtn.disabled = false;
                createTargetBtn.disabled = false;
                updateRectBtn.disabled = false;
                searchBtn.disabled = false;
            }).catch((error) => {
                console.error('Error during search:', error);
                alert('An error occurred during the search.');
                generateBtn.disabled = false;
                createTargetBtn.disabled = false;
                updateRectBtn.disabled = false;
                searchBtn.disabled = false;
            });
        });

        // Helper function to get the current target rectangle
        function getCurrentTargetRect() {
            const targetDiv = document.getElementById('target-rect');
            if (targetDiv.style.display === 'none') return null;

            const left = parseFloat(targetDiv.style.left);
            const top = parseFloat(targetDiv.style.top);
            const width = parseFloat(targetDiv.style.width);
            const height = parseFloat(targetDiv.style.height);

            if (isNaN(left) || isNaN(top) || isNaN(width) || isNaN(height)) {
                return null;
            }

            return { x: left, y: top, width, height };
        }
    </script>

</body>
</html>
```
modify this such that target rect is not fixed viewport position but fixed page position (uses absolute positioning and account for body margins which actually offset the position. say margin left is 10px, then if x is 20px, the resultant x is x+10px because of the margin)