# Complexity

This analysis is on the computing and memory complexity of specific procedures. It provides a rough estimate of operations performed in general and especially focusing on DB reads and writes. It is also an attempt to estimate the memory consumption at its peak.

The primary goal is to come up with decent pricing for functions that can be invoked by a user (via extrinsics) or by untrusted code that prevents DoS attacks.

# Sandboxing

It makes sense to describe the sandboxing module first because the smart-contract module is built upon it.

## Memory

### set

Copies data from the supervisor's memory to the guest's memory.

**complexity**: It doesn't allocate, and the computational complexity is proportional to the number of bytes to copy.

### get

Copies data from the guest's memory to the supervisor's memory.

**complexity**: It doesn't allocate, and the computational complexity is proportional to the number of bytes to copy.

## Instance

### Instantiation

Instantiation of a sandbox module consists of the following steps:

1. Loading the wasm module in the in-memory representation,
2. Performing validation of the wasm code,
3. Setting up the environment which will be used to instantiate the module,
4. Performing the standard wasm instantiation process, which includes (but is not limited to):
    1. Allocating of memory requested by the instance,
    2. Copying static data from the module to newly allocated memory,
    3. Executing the `start` function.

**Note** that the `start` function can be viewed as a normal function and can do anything that a normal function can do, including allocation of more memory or calling the host environment. The complexity of running the `start` function should be considered separately.

In order to start the process of instantiation, the supervisor should provide the wasm module code being instantiated and the environment definition (a set of functions, memories (and maybe globals and tables in the future) available for import by the guest module) for that module. While the environment definition typically is of the constant size (unless mechanisms like dynamic linking are used), the size of wasm is not.

Validation and instantiation in WebAssembly are designed to be able to be performed in linear time. The allocation and computational complexity of loading a wasm module depend on the underlying wasm VM being used. For example, for JIT compilers it can and probably will be non-linear because of compilation. However, for wasmi, it should be linear. We can try to use other VMs that are able to compile code with memory and time consumption proportional to the size of the code.

Since the module itself requests memory, the amount of allocation depends on the module code itself. If untrusted code is being instantiated, it's up to the supervisor to limit the amount of memory available to allocate.

**complexity**: The computational complexity is proportional to the size of wasm code. Memory complexity is proportional to the size of wasm code and the amount of memory requested by the module.

### Preparation to invoke

Invocation of an exported function in the sandboxed module consists of the following steps:

1. Marshalling, copying and unmarshalling the arguments when passing them between the supervisor and executor,
2. Calling into the underlying VM,
3. Marshalling, copying and unmarshalling the result when passing it between the executor and supervisor,

**Note** that the complexity of running the function code itself should be considered separately.

The actual complexity of invocation depends on the underlying VM. Wasmi will reserve a relatively large chunk of memory for the stack before execution of the code, although it's of constant size.

The size of the arguments and the return value depends on the exact function in question, but can be considered as constant.

**complexity**: Memory and computational complexity can be considered as a constant.

### Call from the guest to the supervisor

The executor handles each call from the guest. The execution of it consists of the following steps:

1. Marshalling, copying and unmarshalling the arguments when passing them between the guest and executor,
2. Calling into the supervisor,
3. Marshaling, copying and unmarshalling the result when passing it between the executor and guest.

**Note** that the complexity of running the supervisor handler should be considered separately.

Because calling into the supervisor requires invoking a wasm VM, the actual complexity of invocation depends on the actual VM used for the runtime/supervisor. Wasmi will reserve a relatively large chunk of memory for the stack before execution of the code, although it's of constant size.

The size of the arguments and the return value depends on the exact function in question, but can be considered as a constant.

**complexity**: Memory and computational complexity can be considered as a constant.

# `AccountDb`

`AccountDb` is an abstraction that supports collecting changes to accounts with the ability to efficiently reverting them. Contract
execution contexts operate on the AccountDb. All changes are flushed into underlying storage only after origin transaction succeeds.

Today `AccountDb` is implemented as a cascade of overlays with the direct storage at the bottom. Each overlay is represented by a `Map`. On a commit from an overlay to an overlay, maps are merged. On commit from an overlay to the bottommost `AccountDb` all changes are flushed to the storage. On revert, the overlay is just discarded.

## get_storage, get_code, get_balance

These functions check the local cache for a requested value and, if it is there, the value is returned. Otherwise, these functions will ask an underlying `AccountDb`  for the value. This means that the number of lookups is proportional to the depth of the overlay cascade. If the value can't be found before reaching the bottommost `AccountDb`, then a DB read will be performed (in case `get_balance` the function `free_balance` will be invoked).

A lookup in the local cache consists of at least one `Map` lookup, for locating the specific account. For `get_storage` there is a second lookup: because account's storage is implemented as a nested map, another lookup is required for fetching a storage value by a key.

These functions return an owned value as its result, so memory usage depends on the value being returned.

**complexity**: The memory complexity is proportional to the size of the value. The computational complexity is proportional to the depth of the overlay cascade and the size of the value; the cost is dominated by the DB read though.

## set_storage, set_code, set_balance

These functions only modify the local `Map`.

A lookup in the local cache consists of at least one `Map` lookup, for locating the specific account. For `get_storage` there is a second lookup: because account's storage is implemented as a nested map, another lookup is required for fetching a storage value by a key.

While these functions only modify the local `Map`, if changes made by them are committed to the bottommost `AccountDb`, each changed entry in the `Map` will require a DB write. Moreover, if the balance of the account is changed to be below `existential_deposit` then that account along with all its storage will be removed, which requires time proportional to the number of storage entries that account has. It should be ensured that pricing accounts for these facts.

**complexity**: Each lookup has a logarithmical computing time to the number of already inserted entries. No additional memory is required.

## commit

In this function, all cached values will be inserted into the underlying `AccountDb` or into the storage.

We are doing `N` inserts into `Map` (`O(log M)` complexity) or into the storage, where `N` is the size of the committed `Map` and `M` is the size of the map of the underlying overlay. Consider adjusting the price of modifying the `AccountDb` to account for this (since pricing for the count of entries in `commit` will make the price of commit way less predictable). No additional memory is required.

Note that in case of storage modification we need to construct a key in the underlying storage. In order to do that we need:

- perform `twox_128` hashing over a concatenation of some prefix literal and the `AccountId` of the storage owner.
- then perform `blake2_256` hashing of the storage key.
- concatenation of these hashes will constitute the key in the underlying storage.

There is also a special case to think of: if the balance of some account goes below `existential_deposit`, then all storage entries of that account will be erased, which requires time proprotional to the number of storage entries that account has.

**complexity**: `N` inserts into a `Map` or eventually into the storage (if committed). Every deleted account will induce removal of all its storage which is proportional to the number of storage entries that account has.

## revert

Consists of dropping (in the Rust sense) of the `AccountDb`.

**complexity**: Computing complexity is proportional to a number of changed entries in a overlay. No additional memory is required.

# Executive

## Transfer

This function performs the following steps:

1. Querying source and destination balances from an overlay (see `get_balance`),
2. Querying `existential_deposit`.
3. Executing `ensure_account_liquid` hook.
4. Updating source and destination balance in the overlay (see `set_balance`).

**Note** that the complexity of executing `ensure_account_liquid` hook should be considered separately.

In the course of the execution this function can perform up to 2 DB reads to `get_balance` of source and destination accounts. It can also induce up to 2 DB writes via `set_balance` if flushed to the storage.

Moreover, if the source balance goes below `existential_deposit` then the account will be deleted along with all its storage which requires time proportional to the number of storage entries of that account.

Assuming marshaled size of a balance value is of the constant size we can neglect its effect on the performance.

**complexity**: up to 2 DB reads and up to 2 DB writes (if flushed to the storage) in the standard case. If removal of the source account takes place then it will additionally perform a DB write per one storage entry that the account has. For the current `AccountDb` implementation computing complexity also depends on the depth of the `AccountDb` cascade. Memorywise it can be assumed to be constant.

## Call

This function receives input data for the contract execution. The execution consists of the following steps:

1. Loading code from the DB.
2. `transfer`-ing funds between the caller and the destination account.
3. Executing the code of the destination account.
4. Committing overlayed changed to the underlying `AccountDb`.

**Note** that the complexity of executing the contract code should be considered separately.

Loading code most probably will trigger a DB read, since the code is immutable and therefore will not get into the cache (unless a suicide removes it).

Also, `transfer` can make up to 2 DB reads and up to 2 DB writes (if flushed to the storage) in the standard case. If removal of the source account takes place then it will additionally perform a DB write per one storage entry that the account has.

Finally, all changes are `commit`-ted into the underlying overlay. The complexity of this depends on the number of changes performed by the code. Thus, the pricing of storage modification should account for that.

**complexity**: Up to 3 DB reads. DB read of the code is of dynamic size. There can also be up to 2 DB writes (if flushed to the storage). Additionally, if the source account removal takes place a DB write will be performed per one storage entry that the account has.

## Create

This function takes the code of the constructor and input data. Creation of a contract consists of the following steps:

1. Calling `DetermineContractAddress` hook to determine an address for the contract,
2. `transfer`-ing funds between self and the newly created contract.
3. Executing the constructor code. This will yield the final code of the code.
4. Storing the code for the newly created contract in the overlay.
5. Committing overlayed changed to the underlying `AccountDb`.

**Note** that the complexity of executing the constructor code should be considered separately.

**Note** that the complexity of `DetermineContractAddress` hook should be considered separately as well. Most probably it will use some kind of hashing over the code of the constructor and input data. The default `SimpleAddressDeterminator` does precisely that.

**Note** that the constructor returns code in the owned form and it's obtained via return facilities, which should have take fee for the return value.

Also, `transfer` can make up to 2 DB reads and up to 2 DB writes (if flushed to the storage) in the standard case. If removal of the source account takes place then it will additionally perform a DB write per one storage entry that the account has.

Storing the code in the overlay may induce another DB write (if flushed to the storage) with the size proportional to the size of the constructor code.

Finally, all changes are `commit`-ted into the underlying overlay. The complexity of this depends on the number of changes performed by the constructor code. Thus, the pricing of storage modification should account for that.

**complexity**: Up to 2 DB reads and induces up to 3 DB writes (if flushed to the storage), one of which is dependent on the size of the code. Additionally, if the source account removal takes place a DB write will be performed per one storage entry that the account has.

# Externalities

Each external function invoked from a contract can involve some overhead.

## ext_gas

**complexity**: This is of constant complexity.

## ext_set_storage

This function receives a `key` and `value` as arguments. It consists of the following steps:

1. Reading the sandbox memory for `key` and `value` (see sandboxing memory get).
2. Setting the storage by the given `key` with the given `value` (see `set_storage`).

**complexity**: Complexity is proportional to the size of the `value`. This function induces a DB write of size proportional to the `value` size (if flushed to the storage), so should be priced accordingly.

## ext_get_storage

This function receives a `key` as an argument. It consists of the following steps:

1. Reading the sandbox memory for `key` (see sandboxing memory get).
2. Reading the storage with the given key (see `get_storage`). It receives back the owned result buffer.
3. Replacing the scratch buffer.

Key is of a constant size. Therefore, the sandbox memory load can be considered to be of constant complexity.

However, a read from the contract's storage can hit the DB, and the size of this read is dynamical.

**complexity**: The memory and computing complexity is proportional to the size of the fetched value.

## ext_call

This function receives the following arguments:

- `callee` buffer of a marshaled `AccountId`,
- `gas` limit which is plain u64,
- `value` buffer of a marshaled `Balance`,
- `input_data`. An arbitrarily sized byte vector.

It consists of the following steps:

1. Loading `callee` buffer from the sandbox memory (see sandboxing memory get) and then decoding it.
2. Loading `value` buffer from the sandbox memory and then decoding it.
3. Loading `input_data` buffer from the sandbox memory.
4. Invoking `call` executive function.

Loading of `callee` and `value` buffers should be charged. This is because the sizes of buffers are specified by the calling code, even though marshaled representations are, essentially, of constant size. This can be fixed by assigning an upper bound for sizes of `AccountId` and `Balance`.

Loading `input_data` should be charged in any case.

**complexity**: All complexity comes from loading buffers and executing `call` executive function. The former component is proportional to the sizes of `callee`, `value` and `input_data` buffers. The latter component completely depends on the complexity of `call` executive function, and also dominated by it.

## ext_create

This function receives the following arguments:

- `init_code`, a buffer which contains the code of the constructor.
- `gas` limit which is plain u64
- `value` buffer of a marshaled `Balance`
- `input_data`. an arbitrarily sized byte vector.

It consists of the following steps:

1. Loading `init_code` buffer from the sandbox memory (see sandboxing memory get) and then decoding it.
2. Loading `value` buffer from the sandbox memory and then decoding it.
3. Loading `input_data` buffer from the sandbox memory.
4. Invoking `create` executive function.

Loading of `value` buffer should be charged. This is because the size of the buffer is specified by the calling code, even though marshaled representation is, essentially, of constant size. This can be fixed by assigning an upper bound for size for `Balance`.

Loading `init_code` and `input_data` should be charged in any case.

**complexity**: All complexity comes from loading buffers and executing `create` executive function. The former component is proportional to the sizes of `init_code`, `value` and `input_data` buffers. The latter component completely depends on the complexity of `create` executive function and also dominated by it.

## ext_return

This function receives a `data` buffer as an argument. Execution of the function consists of the following steps:

1. Loading `data` buffer from the sandbox memory (see sandboxing memory get),
2. Trapping

**complexity**: The complexity of this function is proportional to the size of the `data` buffer.

## ext_input_size

**complexity**: This function is of constant complexity.

## ext_input_copy

This function copies slice of data from the input buffer to the sandbox memory. The calling code specifies the slice length. Execution of the function consists of the following steps:

1. Storing a specified slice of the input data into the sandbox memory (see sandboxing memory set)

**complexity**: The computing complexity of this function is proportional to the length of the slice. No additional memory is required.

## ext_scratch_size

**complexity**: This function is of constant complexity.

## ext_scratch_copy

This function copies slice of data from the scratch buffer to the sandbox memory. The calling code specifies the slice length. Execution of the function consists of the following steps:

1. Storing a specified slice of the scratch buffer into the sandbox memory (see sandboxing memory set)

**complexity**: The computing complexity of this function is proportional to the length of the slice. No additional memory is required.
