| Taskflow: A General-purpose Task-parallel Programming System

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Parallel Merge

Taskflow provides a function for constructing a task to merge two sorted ranges into a single sorted output range in parallel.

# Include the Header

You need to include the header file, `taskflow/algorithm/merge.hpp`, for creating a parallel-merge task.

#include <taskflow/algorithm/merge.hpp>

# Motivation

The standard library `std::merge` walks both input sequences simultaneously from left to right in a single thread, producing a sorted output in O(n+m) time. While optimal for a single core, this sequential walk cannot exploit multiple CPU cores — at any point only one comparison is in flight. For large inputs (e.g. merging two sorted arrays of tens of millions of elements), this leaves the majority of available hardware idle.

Taskflow's parallel merge divides the output into `W` independent chunks (one per worker thread) and uses the co-rank technique to identify each worker's exact input sub-ranges, allowing all workers to merge their chunks simultaneously with no synchronization.

# Create a Parallel-Merge Task

The task created by tf::Taskflow::merge(B1 first1, E1 last1, B2 first2, E2 last2, O d_first) merges the two sorted ranges `[first1, last1)` and `[first2, last2)` into the output range beginning at `d_first`, using `std::less` as the comparator. It represents the parallel execution of `std::merge:`

std::merge(first1, last1, first2, last2, d_first);

The following example merges two sorted vectors of one million integers using four worker threads:

#include <taskflow/algorithm/merge.hpp>
tf::Taskflow taskflow;
tf::Executor executor(4);
std::vector<int> seq1(1000000), seq2(1000000), output(2000000);
// fill and sort seq1 and seq2 ...
taskflow.merge(
seq1.begin(), seq1.end(),
seq2.begin(), seq2.end(),
output.begin()
);
executor.run(taskflow).wait();
// output is now the sorted merge of seq1 and seq2
tf::Executor
class to create an executor
Definition executor.hpp:62
tf::FlowBuilder::merge
Task merge(B1 first1, E1 last1, B2 first2, E2 last2, O d_first)
merges two sorted ranges into a single sorted output using the std::less comparator
tf::Taskflow
class to create a taskflow object
Definition taskflow.hpp:64

To merge with a custom comparator, pass it as the last argument. Both input ranges must be sorted with respect to that comparator:

// descending order
std::sort(seq1.begin(), seq1.end(), std::greater<int>{});
std::sort(seq2.begin(), seq2.end(), std::greater<int>{});
taskflow.merge(
seq1.begin(), seq1.end(),
seq2.begin(), seq2.end(),
output.begin(),
std::greater<int>{}
);
executor.run(taskflow).wait();

Note
  Both input ranges must be sorted with respect to the comparator before the merge task runs. Passing unsorted input is undefined behavior. The output range must not overlap either input range. Both input iterators must be random-access iterators.

# Capture Iterators by Reference

You can pass iterators by reference using std::ref to marshal parameter updates between dependent tasks. This is useful when the ranges are not known at task-graph construction time but are initialized by an upstream task.

std::vector<int> seq1, seq2, output;
std::vector<int>::iterator beg1, end1, beg2, end2, d_beg;
tf::Task init = taskflow.emplace([&]() {
// fill and sort seq1, seq2 at runtime
beg1 = seq1.begin(); end1 = seq1.end();
beg2 = seq2.begin(); end2 = seq2.end();
d_beg = output.begin();
});
tf::Task merge_task = taskflow.merge(
std::ref(beg1), std::ref(end1),
std::ref(beg2), std::ref(end2),
std::ref(d_beg)
);
// wrong! iterators are captured by copy at construction time
// tf::Task merge_task = taskflow.merge(beg1, end1, beg2, end2, d_beg);
init.precede(merge_task);
executor.run(taskflow).wait();
tf::FlowBuilder::emplace
Task emplace(C &&callable)
creates a static task
Definition flow_builder.hpp:1738
tf::Task
class to create a task handle over a taskflow node
Definition task.hpp:569
tf::Task::precede
Task & precede(Ts &&... tasks)
adds precedence links from this to other tasks
Definition task.hpp:1258

When `init` finishes, the merge task reads the updated iterators and merges the two runtime-defined sequences.

# Understanding the Parallel Merge Algorithm

## Step 1: Flat Output Partitioning

The merged output has `N` = n + m elements (where`n = |seq1|` and `m = |seq2|`).
The algorithm divides the output into `W` equal contiguous chunks of size `K = N/W`, one per worker thread. Worker `w` is responsible for writing output positions `[w*K, (w+1)*K)`. The key challenge here is that for each output chunk, which elements of `seq1` and `seq2` belong to it? This is what the co-rank function solves.

## Step 2: Co-rank to Identify Input Subranges

For a given output position `rank`, `co_rank(rank)` finds `i` and `j` such that:

- The first `rank` elements of the merged output consist of exactly `i` elements from `seq1`[0..i) and `j` elements from `seq2`[0..j).
- These `i` + `j` elements are interleaved in sorted order in the output. Note that `co_rank` does not decide how they go in blocks but only identifies how many elements come from each sequence. The actual placement of these elements are accomplished via std::merge.

Once a worker knows `(i_start, j_start)` at its chunk's beginning and `(i_end, j_end)` at its chunk's end, it can independently merge `seq1[i_start..i_end)` with `seq2[j_start..j_end)` and write the result directly to its output region, with no communication with other workers.

## Step 3: How co_rank works? Binary Search!

For a given `rank`, co_rank binary-searches for the unique `i` in the range `[max(0, rank-m), min(n, rank)]` such that the partition `(seq1[0..i), seq2[0..rank-i))` is valid. A partition is valid when:

- The last element of seq1's slice does not exceed the first unused element of `seq2`: `seq1[i-1] <= seq2[j]`
- The last element of seq2's slice does not exceed the first unused element of `seq1`: `seq2[j-1] <= seq1[i]`

Because both sequences are sorted, as `i` increases:

- `seq1[i-1]` increases (moving right in `seq1`)
- `seq2[j-1]` decreases (moving left in `seq2`, since `j = rank - i` decreases)

This means the condition `seq2`[j-1] <= `seq1`[i] transitions from `false` to `true` exactly once — making binary search applicable. The algorithm needs to check only one condition per iteration:

- If `seq2`[j-1] > `seq1`[i]: `i` is too large -> `high` = i
- Otherwise: `i` is not large enough -> `low` = i + 1

The figure below shows two iterations of the binary search for `rank=5` on sequences `seq1=[1,3,5,7,9,11]` and `seq2=[2,4,6,8,10,12]`:

The search converges to `i=3`, `j=2`: the first 5 merged elements consist of `seq1[0,3)=[1,3,5]` and `seq2[0,2)=[2,4]`, which interleave to `[1,2,3,4,5]`.

## Why parallel merge uses no partitioner

Unlike tf::Taskflow::for_each, where users can configure a partitioner to adapt to unequal per-element costs, `std::merge` on a chunk of size `K` always costs `O(K)` regardless of the data values. There is little load imbalance to mitigate. As a result, tf::Taskflow::merge always adopts static partitioning, i.e., `W` chunks of size `N/W` and one per worker. which is always the optimal strategy for parallel merge.